Illinois ABE/ASE
Mathematics
Content Standards
Dr. Karen Hunter Anderson
Executive Director
Illinois Community College Board
Jennifer K. Foster
Associate Vice President for Adult Education and Workforce Development
Illinois Community College Board
June 2012
REVISED (May 2014)
For the purpose of compliance with Public Law 101-166 (The Stevens Amendment),
approximately 100% federal funds were used to produce this document.
Table of Contents
Acknowledgements ......................................................................................................... 4
Foreword ......................................................................................................................... 5
Introduction to the Mathematics Standards ................................................................... 10
Understanding Mathematics ...................................................................................... 12
Illinois ABE/ASE Standards for Mathematical Practice .............................................. 13
Connecting the Standards for Mathematical Practice to the Standards for
Mathematical Content ................................................................................................ 16
Guiding Principles for Mathematics Programs ........................................................... 17
How to Read the Six NRS Level Standards ............................................................... 23
The Standards for Mathematical Content ...................................................................... 25
NRS Level 1 Overview............................................................................................... 26
NRS Level 2 Overview............................................................................................... 36
NRS Level 3 Overview............................................................................................... 49
NRS Level 4 Overview............................................................................................... 67
NRS Level 5 Overview............................................................................................... 95
NRS Level 6 Overview............................................................................................. 121
Math Glossary ............................................................................................................. 137
Tables and Illustrations of Key Mathematical Properties, Rules, and Number Sets .... 159
Acknowledgements
The Adult Education and Family Literacy Program of the Illinois Community College
Board (ICCB) recognizes the Adult Basic Education (ABE) / Adult Secondary Education
(ASE) educators who contributed to the development of the Illinois ABE/ASE Content
Standards. The dedication, commitment, and hard work of administrators, coordinators,
and instructors created this document which reflects the knowledge of practitioners in
Illinois programs.
The Math Team aligned the previous version of the Illinois ABE/ASE Content Standards
with the Common Core State Standards through a process of research, discussion, and
revision. The membership included:
Joshua Anderson
Waukegan Public Library
Dr. Sharon Bryant
City Colleges of Chicago
Kelly Gagnon
Richland Community
College
Dr. Tania Giordani
College of Lake County
Mark Harrison
Urbana Adult Education
Elizabeth Hobson
Elgin Community
College
Emilie McCallister
Joliet Junior College
Libby Serkies
Central Illinois Adult
Education Service Center
The ICCB would like to thank the following individuals and organizations for their
leadership and assistance throughout the project:



Dr. Akemi Haynie, Math Team Leader
Dawn Hughes, Project Leader
Central Illinois Adult Education Service Center (CIAESC)
Foreword
What Are Content Standards?
Content standards describe what learners should know and be able to do in a specific
content area. The Illinois ABE/ASE Content Standards broadly define what learners
who are studying reading, writing, and math should know and be able to do as a result
of ABE/ASE instruction at a particular level. Content standards also help teachers
ensure their students have the skills and knowledge they need to be successful by
providing clear goals for student learning.
The Illinois ABE/ASE Content Standards should be used as a basis for curriculum
design and may also be used to assist programs and teachers with selecting or
designing appropriate instructional materials, instructional techniques, and ongoing
assessment strategies. Standards do not tell teachers how to teach, but they do help
teachers figure out the knowledge and skills their students should have so that teachers
can build the best lessons and environments for their classrooms.
Why Are the Illinois ABE/ASE Content Standards Necessary?
The Illinois ABE/ASE Content Standards serve multiple purposes. They:



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Provide a common language for ABE/ASE levels among programs;
Assist programs with ABE/ASE curriculum development;
Provide guidance for new ABE/ASE instructors; and
Ensure quality instruction through professional development.
Provide a Common Language for ABE/ASE Levels
ABE/ASE classes are very different across Illinois programs. The Illinois ABE/ASE
Content Standards provide a description of what students should learn at each National
Reporting System (NRS) level so that adult education practitioners have a common
language when discussing ABE/ASE levels. Having a common language among levels
and programs will help ABE/ASE learners who move from level to level within the same
program or who move from one ABE/ASE program to another.
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We need standards to ensure that all students, no matter where they live in the state of
Illinois, are prepared for success in postsecondary education and/or the workforce. The
Illinois ABE/ASE Content Standards will help ensure that our students are receiving a
consistent education from program to program across the state. These standards will
provide a greater opportunity to share experiences and best practices within and across
the state that will improve our ability to best serve the needs of our students.
Assist Programs with ABE/ASE Curriculum Development
The Illinois ABE/ASE Content Standards should serve as the basis for a program’s
curriculum development process. For programs with an existing curriculum, that
curriculum should be aligned to the standards. For programs without a curriculum, the
standards provide an excellent framework and starting point for the curriculum
development process.
Provide Guidance for New ABE/ASE Instructors
The Illinois ABE/ASE Content Standards provide guidance for new instructors who may
have limited training in teaching adults enrolled in adult basic classes. The standards
serve as a basis for what they should teach and include in their lesson plans.
Ensure Quality Instruction through Professional Development
In order to implement the Illinois ABE/ASE Content Standards, program staff
(administrators and instructors) will participate in professional development on
implementation of the standards. These professional development sessions will address
curriculum design, instructional materials, instructional techniques, and ongoing
assessment strategies related to the standards. They will also provide an excellent
opportunity for new and experienced ABE/ASE instructors to develop and refine their
teaching skills.
Why Were the Illinois ABE/ASE Content Standards Revised?
The GED® 21st Century Initiative will introduce a new assessment to our students in
January 2014. The GED® 21st Century Initiative is committed to helping more adults
become career- and college-ready by transforming the GED® test into a comprehensive
program. By building a more robust assessment, complete with preparation tools and
transitions to college and careers, GED® Testing Service and ACE hope to increase the
number of adults who can enter and succeed in college and the workforce. The new
assessment will be closely aligned with the Common Core State Standards and will be
administered through computer-based testing (CBT), although paper-based testing
(PBT) will still be available under certain circumstances or as an accommodation.
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The Common Core State Standards Initiative is a state-led effort to establish a shared
set of clear educational standards for English language arts and mathematics that
states can voluntarily adopt. The standards have been informed by the best available
evidence and the highest state standards across the country and globe and designed by
a diverse group of teachers, experts, parents, and school administrators. These
standards are designed to ensure that students graduating from high school are
prepared to go to college or enter the workforce. The standards are benchmarked to
international standards to guarantee that our students are competitive in the emerging
global marketplace. The Illinois State Board of Education adopted the Common Core
State Standards in June 2010.
In April 2013, the Office of Vocational and Adult Education (OVAE) released the highlyanticipated report, College and Career Readiness (CCR) Standards for Adult
Education1. The report was the result of a nine-month process that examined the
Common Core State Standards from the perspective of adult education. It was funded
to provide a set of manageable yet significant CCR standards that reflect broad
agreement among subject matter experts in adult education about what is desirable for
adult students to know to be prepared for the rigors of postsecondary education and
training.
How Were the Illinois ABE/ASE Content Standards Revised?
The original Illinois ABE/ASE Content Standards and Benchmarks (April 2011) were the
result of several federal and state initiatives that addressed the need for content
standards in adult education programs. During September 2011 a statewide application
process was opened to adult education teachers, administrators, transition coordinators,
etc., in order to participate in the ABE/ASE Content Standard Project. Selected
applicants were assigned to either the math, reading, or writing team and began work in
November 2011. The task for each team was to align the original Illinois ABE/ASE
Content Standards and Benchmarks (April 2011) with the Common Core State
Standards, College Readiness Standards, Career Clusters Essential Knowledge and
Skills, Evidence Based Reading Instruction, the International Society for Technology in
Education’s National Educational Technology Standards for Students, and other
standards to ensure student success in post-secondary education and/or employment.
The teams spent over six months reviewing, aligning, and editing the ABE/ASE content
standards. A draft was submitted to the Illinois Community College Board (ICCB) in
1
College and Career Readiness (CCR) for Adult Education, 2013, www.lincs.ed.gov/publications/pdf/CCRStandardsAdultEd.pdf.
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February 2012. Select content area experts reviewed the draft in April 2012, and the
standards went through an open comment period in May 2012.
After the release of the College and Career Readiness (CCR) Standards for Adult
Education in April 2013, the Illinois ABE/ASE Content Standards that were published in
June 2012 were reexamined. Because the content standards were already aligned with
the Common Core State Standards, very few revisions were necessary. Additions have
been made to the Reference column to highlight the CCR Standards that have been
identified by OCTAE (Office of Career, Technical, and Adult Education – formerly known
as OVAE). Furthermore, a gap analysis of the Illinois ABE/ASE Content Standards
(June 2012) with the CCR Standards for Adult Education was completed by federal
representatives in April 2014. The gap analysis examined the degree of alignment
between the Illinois ABE/ASE Content Standards and key advance in the CCR
Standards for Adult Education. The gap analysis concluded that “the standards – as
they are presently composed – have many strengths, particularly the organization and
structure of the standards document.”
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Design of the Illinois ABE/ASE Content Standards
Adult education programs nationwide use the NRS educational functioning levels to
provide information to the federal government about student progress. This uniform
implementation makes it possible to compare data across programs. The Illinois
ABE/ASE content standards conform to the NRS structure for consistency and
accountability. There are six NRS educational functioning levels from beginning literacy
and adult basic education through adult secondary education. The six levels each have
titles and are identified by grade equivalency:
NRS Educational Functioning Levels
1 Beginning ABE Literacy
2 Beginning Basic Education
3 Low Intermediate Basic Education
4 High Intermediate Basic Education
5 Low Adult Secondary Education
6 High Adult Secondary Education
Grade Level Equivalency
0-1.9
2-3.9
4-5.9
6-8.9
9-10.9
11.0-12.9
Content standards for reading, writing, speaking, listening, and math skills are included
in this document. The essential knowledge and skills statements from the Career
Clusters have also been incorporated into the content standards. The use of technology
has been infused in this document as well. We would also like to reinforce that because
these standards have been aligned to the Common Core State Standards, we are
ensuring that our students are college and career ready. These standards are by no
means meant to limit a teacher’s creativity. Certainly some of the best teaching is done
across the curriculum including some or all of the subject areas.
Assessment
Ongoing assessment of the Illinois ABE/ASE Content Standards should be a part of
every lesson. Learners can demonstrate their mastery of a particular standard through
ongoing assessment strategies such as demonstrations, project-based learning,
presentations, simulation, out-of-class activities, and other nontraditional assessment
strategies. Ongoing assessment is an integral part of instruction in standards-based
education.
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Introduction to the Mathematics Standards2
The Illinois ABE/ASE Math Common Core State Standards provide a consistent, clear
understanding of what adult learners are expected to learn, so adult educators know
what they need to do to help them. The standards are designed to be robust and
relevant to the real world, reflecting the knowledge and skills that our students need for
success in college and careers.
For over a decade, research studies of mathematics education in high-performing
countries have pointed to the conclusion that the mathematics curriculum in the United
States must become substantially more focused and coherent in order to improve
mathematics achievement in this country. To deliver on the promise of common
standards, the standards must address the problem of a curriculum that is “a mile wide
and an inch deep.” These standards are a substantial answer to that challenge.
It is important to recognize that “fewer standards” are no substitute for focused
standards. Achieving “fewer standards” would be easy to do by resorting to broad,
general statements. Instead, these standards aim for clarity and specificity.
William Schmidt and Richard Houang (2002) have said that content standards and
curricula are coherent if they are:
Articulated over time as a sequence of topics and performances that are logical
and reflect, where appropriate, the sequential or hierarchical nature of the
disciplinary content from which the subject matter derives. That is, what and how
students are taught should reflect not only the topics that fall within a certain
academic discipline, but also the key ideas that determine how knowledge is
organized and generated within that discipline. This implies that “to be coherent,”
a set of content standards must evolve from particulars (e.g., the meaning and
operations of whole numbers, including simple math facts and routine
computational procedures associated with whole numbers and fractions) to
deeper structures inherent in the discipline. These deeper structures then serve
as a means for connecting the particulars (such as an understanding of the
rational number system and its properties).
These standards endeavor to follow such a design, not only by stressing conceptual
understanding of key ideas, but also by continually returning to organizing principles
such as place value or the laws of arithmetic to structure those ideas.
2
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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In addition, the “sequence of topics and performances” that is outlined in a body of
mathematics standards must also respect what is known about how students learn. As
Confrey (2007) points out, developing “sequenced obstacles and challenges for
students…absent the insights about meaning that derive from careful study of learning,
would be unfortunate and unwise.” In recognition of this, the development of these
standards began with research-based learning progressions detailing what is known
today about how students’ mathematical knowledge, skill, and understanding develop
over time.
Adult learners need to envision mathematics as part of their daily language, a tool, and
an art form with which they can communicate ideas, solve problems, and explore the
world around them. When adult educators integrate technology and the Illinois State
Career Clusters in their delivery of instruction, learners will be able to connect to, use,
and express mathematical ideas in multiple ways in real life situations. Learners will
also make, refine, and explore conjectures on the basis of evidence and use a variety of
reasoning and proof techniques to confirm or disprove those conjectures. Through this
process, they will develop mathematical reasoning skills and become adept at
evaluating their own thinking.
Adult learners will have an understanding of how numbers are used and represented.
They will be able to estimate and use basic operations to both solve everyday problems
and confront more involved calculations in algebraic and statistical settings. They will
be able to read, write, visualize, and talk about ways in which mathematical problems
can be solved in both theoretical and practical situations. They will be able to
communicate relationships in geometric and statistical settings through drawings and
graphs.
Finally, as adult learners transition from level to level, they will demonstrate an
understanding and use a variety of resources and tools to communicate their thinking
from situation to situation and achieve additional academic knowledge and skills
required to pursue a full range of career and postsecondary education opportunities.
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Understanding Mathematics3
These standards define what ABE/ASE adult learners should understand and be able to
do in their study of mathematics. Asking a student to understand something means
asking an adult educator to assess whether the adult learner has understood it. But
what does mathematical understanding look like? One hallmark of mathematical
understanding is the ability to justify, in a way appropriate to the student’s mathematical
maturity, why a particular mathematical statement is true or where a mathematical rule
comes from. There is a world of difference between a student who can summon a
mnemonic device to expand a product such as (a + b)(x + y) and a student who can
explain where the mnemonic comes from. The student who can explain the rule
understands the mathematics, and may have a better chance to succeed at a less
familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and
procedural skill are equally important, and both are assessable using mathematical
tasks of sufficient richness.
The standards are level specific but do not define the intervention methods or materials
necessary to support adult learners who are well below or well above level
expectations. It is also beyond the scope of the Standards to define the full range of
supports appropriate for English language learners (ELL) and for students with special
needs. At the same time, all students must have the opportunity to learn and meet the
same high standards if they are to access the knowledge and skills necessary to
succeed and transition to college and careers in their post-school lives. The standards
should be read as allowing for the widest possible range of adult learners to participate
fully from the outset, along with appropriate accommodations to ensure maximum
participation of students with special education needs.
ELL students are capable of participating in mathematical discussions as they learn
English. Mathematics instruction for ELL students should draw on multiple resources
and modes available in classrooms---such as objects, drawings, inscriptions, and
gestures---as well as mathematical experiences outside of the classroom.
Promoting a culture of high expectations for all students is a fundamental goal of the
Illinois ABE/ASE State Standards. In order to participate with success in the general
curriculum, both ELL and students with disabilities should be provided the necessary
supports, accommodations, and services where applicable.
No set of level-specific standards can fully reflect the great variety in abilities, needs,
learning rates, and achievement levels of adult learners in any given classroom.
3
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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However, the standards do provide clear signposts along the way to the goal of college
and career readiness for all students.
The standards begin here with eight Standards for Mathematical Practice.
Illinois ABE/ASE Standards for Mathematical Practice
The Illinois ABE/ASE Standards for Mathematical Practice describe a variety of
expertise that mathematics adult educators at all levels should seek to develop in their
adult students4. These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education. The first of these are the National
Council of Teachers of Mathematics (NCTM) process standards of problem solving,
reasoning and proof, communication, representation, and connections. The second are
the strands of mathematical proficiency specified in the National Research Council’s
report Adding It Up: adaptive reasoning, strategic competence, conceptual
understanding (comprehension of mathematical concepts, operations and relations),
procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately), and productive disposition (habitual inclination to see mathematics as
sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own
efficacy).
1. Make sense of problems and persevere in solving them.
Mathematically proficient adult learners start by explaining to themselves the meaning
of a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make conjectures about the form and
meaning of the solution and plan a solution pathway rather than simply jumping into a
solution attempt. They consider analogous problems, and try special cases and simpler
forms of the original problem in order to gain insight into its solution. They monitor and
evaluate their progress and change course if necessary. Higher level students might,
depending on the context of the problem, transform algebraic expressions or change
the viewing window on their graphing calculator to get the information they need.
Mathematically proficient adult learners can explain correspondences between
equations, verbal descriptions, tables, and graphs, or draw diagrams of important
features and relationships, graph data, and search for regularity or trends. Lower level
students might rely on using concrete objects or pictures to help conceptualize and
solve a problem. Mathematically proficient students check their answers to problems
4
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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using a different method, and they continually ask themselves, “Does this make sense?”
They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient adult learners make sense of quantities and their relationships
in problem situations. They bring two complementary abilities to bear on problems
involving quantitative relationships: the ability to decontextualize (to abstract a given
situation, represent it symbolically, and manipulate the representing symbols as if they
have a life of their own, without necessarily attending to their referents); and the ability
to contextualize (to pause as needed during the manipulation process in order to probe
into the referents for the symbols involved). Quantitative reasoning entails habits of
creating a coherent representation of the problem at hand; considering the units
involved; attending to the meaning of quantities, not just how to compute them; and
knowing and flexibly using different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient adult learners understand and use stated assumptions,
definitions, and previously established results in constructing arguments. They make
conjectures and build a logical progression of statements to explore the truth of their
conjectures. They are able to analyze situations by breaking them into cases, and can
recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data,
making plausible arguments that take into account the context from which the data
arose. Mathematically proficient adult learners are also able to compare the
effectiveness of two plausible arguments, distinguish correct logic or reasoning from
that which is flawed, and—if there is a flaw in an argument—explain what it is. Lower
level adult learners can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct,
even though they are not generalized or made formal until higher levels. Later, students
learn to determine domains to which an argument applies. Students at all levels can
listen or read the arguments of others, decide whether they make sense, and ask useful
questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient adult learners can apply the mathematics they know to solve
problems arising in everyday life, society, and the workplace. In lower levels, this might
be as simple as writing an addition equation to describe a situation. In intermediate
levels, adult learners might apply proportional reasoning to plan a school event or
analyze a problem in the community. By the upper levels, an adult learner might use
geometry to solve a design problem or use a function to describe how one quantity of
interest depends on another. Mathematically proficient adult learners who can apply
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what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to
identify important quantities in a practical situation and map their relationships using
such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can
analyze those relationships mathematically to draw conclusions. They routinely interpret
their mathematical results in the context of the situation and reflect on whether the
results make sense, possibly improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient adult learners consider the available tools when solving a
mathematical problem. These tools might include pencil and paper, concrete models, a
ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical
package, or dynamic geometry software. Proficient adult students are sufficiently
familiar with tools appropriate for their grade or course to make sound decisions about
when each of these tools might be helpful, recognizing both the insight to be gained and
their limitations. For example, mathematically proficient advanced level adult learners
analyze graphs of functions and solutions generated using a graphing calculator. They
detect possible errors by strategically using estimation and other mathematical
knowledge. When making mathematical models, they know that technology can enable
them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various levels are
able to identify relevant external mathematical resources, such as digital content located
on a website, and use them to pose or solve problems. They are able to use
technological tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient adult learners try to communicate precisely to others. They try
to use clear definitions in discussion with others and in their own reasoning. They state
the meaning of the symbols they choose, including using the equal sign consistently
and appropriately. They are careful about specifying units of measure, and labeling
axes to clarify the correspondence with quantities in a problem. They calculate
accurately and efficiently, expressing numerical answers with a degree of precision
appropriate for the problem context. In the lower levels, adult learners give carefully
formulated explanations to each other. By the time they reach the advanced levels, they
have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient adult learners look closely to discern a pattern or structure.
Lower level students, for example, might notice that three and seven more is the same
amount as seven and three more, or they may sort a collection of shapes according to
how many sides the shapes have. Later, students will see 7 × 8 equals the wellremembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
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the expression x2 + 9x + 14, higher level students can see the 14 as 2 × 7 and the 9 as
2 + 7. They recognize the significance of an existing line in a geometric figure and can
use the strategy of drawing an auxiliary line for solving problems. They also can step
back for an overview and shift perspective. They can see complicated things, such as
some algebraic expressions, as single objects or as being composed of several objects.
For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square
and use that to realize that its value cannot be more than 5 for any real numbers x and
y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient adult learners notice if calculations are repeated and look both
for general methods and for shortcuts. Intermediate level students might notice when
dividing 25 by 11 that they are repeating the same calculations over and over again,
and conclude they have a repeating decimal. By paying attention to the calculation of
slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, intermediate level students might abstract the equation (y – 2)/(x – 1) = 3. Noticing
the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1),
and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a
geometric series. As they work to solve a problem, mathematically proficient adult
students maintain oversight of the process, while attending to the details. They
continually evaluate the reasonableness of their intermediate results.
Connecting the Standards for Mathematical Practice to the
Standards for Mathematical Content5
The Standards for Mathematical Practice describe ways in which developing student
practitioners of the discipline of mathematics increasingly ought to engage with the
subject matter as they grow in mathematical maturity and expertise throughout the
literacy to intermediate and adult secondary levels. All adult educators, instructional
leaders, designers of curricula, assessments, and professional development should all
attend to the need to connect the mathematical practices to mathematical content in
mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and
understanding. Expectations that begin with the word “understand” are often especially
good opportunities to connect the practices to the content. Adult learners who lack
understanding of a topic may rely on procedures too heavily. Without a flexible base
5
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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from which to work, they may be less likely to consider analogous problems, represent
problems coherently, justify conclusions, apply the mathematics to practical situations,
use technology mindfully to work with the mathematics, explain the mathematics
accurately to other students, step back for an overview, or deviate from a known
procedure to find a shortcut. In short, a lack of understanding effectively prevents an
adult learner from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are
potential “points of intersection” between the Standards for Mathematical Content and
the Standards for Mathematical Practice. These points of intersection are intended to be
weighted toward central and generative concepts in the school mathematics curriculum
that most merit the time, resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction, assessment, professional development,
and student achievement in mathematics.
Guiding Principles for Mathematics Programs6
The following six Guiding Principles are philosophical statements that underlie the
Standards for Mathematical Practice, Standards for Mathematical Content, and other
resources in this ABE/ASE curriculum framework. They should guide the construction
and evaluation of mathematics programs in adult education throughout the state of
Illinois. The Standards for Mathematical Practice are interwoven throughout the Guiding
Principles.
Guiding Principle 1: Learning
Mathematical ideas should be explored in ways that stimulate curiosity, create
enjoyment of mathematics, and develop depth of understanding.
Adult learners need to understand mathematics deeply and use it effectively. The
Standards for Mathematical Practice describe ways in which students increasingly
engage with the subject matter as they grow in mathematical maturity and expertise
through the six NRS levels.
To achieve mathematical understanding, adult learners should have a balance of
mathematical procedures and conceptual understanding. Adult learners should be
actively engaged in doing meaningful mathematics, discussing mathematical ideas, and
applying mathematics in interesting, thought-provoking situations. Student
6
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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understanding is further developed through ongoing reflection about cognitively
demanding and worthwhile tasks.
Tasks should be designed to challenge every adult learner in multiple ways. Short- and
long-term investigations that connect procedures and skills with conceptual
understanding are integral components of an effective mathematics program. Activities
should build upon curiosity and prior knowledge, and enable students to solve
progressively deeper, broader, and more sophisticated problems. (See Standard for
Mathematical Practice 1: Make sense of problems and persevere in solving them.)
Mathematical tasks reflecting sound and significant mathematics should generate active
classroom talk, promote the development of conjectures, and lead to an understanding
of the necessity for mathematical reasoning. (See Standard for Mathematical Practice 2:
Reason abstractly and quantitatively.)
Guiding Principle 2: Teaching
An effective mathematics program is based on a carefully designed set of content
standards that are clear and specific, focused, and articulated over time as a
coherent sequence.
The sequence of topics and performances should be based on what is known about
how adult learners’ mathematical knowledge, skill, and understanding develop over
time. What and how students are taught should reflect not only the topics within
mathematics but also the key ideas that determine how knowledge is organized and
generated within mathematics. (See Standard for Mathematical Practice 7: Look for and
make use of structure.) Adult learners should be given the opportunity to apply their
learning and to show their mathematical thinking and understanding. This requires adult
educators who have a deep knowledge of mathematics as a discipline.
Mathematical problem solving is the hallmark of an effective mathematics program. Skill
in mathematical problem solving requires practice with a variety of mathematical
problems as well as a firm grasp of mathematical techniques and their underlying
principles. Armed with this deeper knowledge, the adult learner can then use
mathematics in a flexible way to attack various problems and devise different ways of
solving any particular problem. (See Standard for Mathematical Practice 8: Look for and
express regularity in repeated reasoning.) Mathematical problem solving calls for
reflective thinking, persistence, learning from the ideas of others, and going back over
one's own work with a critical eye. Adult learners should be able to construct viable
arguments and critique the reasoning of others. They should analyze situations and
justify their conclusions, communicate their conclusions to others, and respond to the
arguments of others. (See Standard for Mathematical Practice 3: Construct viable
arguments and critique the reasoning of others.) Adult learners at all levels should be
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able to listen or read the arguments of others, decide whether they make sense, and
ask questions to clarify or improve the arguments.
Mathematical problem solving provides students with experiences to develop other
mathematical practices. Success in solving mathematical problems helps to create an
abiding interest in mathematics. Adult students learn to model with mathematics and to
apply the mathematics that they know to solve problems arising in everyday life, society,
and the workplace. (See Standard for Mathematical Practice 4: Model with
mathematics.)
For a mathematics program to be effective, it must also be taught by knowledgeable
adult educators. According to Liping Ma, “The real mathematical thinking going on in a
classroom, in fact, depends heavily on the teacher's understanding of mathematics.”7 A
landmark study in 1996 found that students with initially comparable academic
achievement levels had vastly different academic outcomes when teachers’ knowledge
of the subject matter differed.8 The message from the research is clear: having
knowledgeable teachers really does matter; teacher expertise in a subject drives
student achievement. “Improving teachers’ content subject matter knowledge and
improving students’ mathematics education are thus interwoven and interdependent
processes that must occur simultaneously.”9
Guiding Principle 3: Technology
An essential tool that should be used strategically in mathematics education.
Technology enhances the mathematics curriculum in many ways. Tools such as
measuring instruments, manipulatives (such as base ten blocks and fraction pieces),
scientific and graphing calculators, and computers with appropriate software, if properly
used, contribute to a rich learning environment for developing and applying
mathematical concepts. However, appropriate use of calculators is essential; calculators
should not be used as a replacement for basic understanding and skills. Adult education
students that are in (levels 1-3) should learn how to perform the basic arithmetic
operations independent of the use of a calculator.10 Although the use of a graphing
calculator can help adult learners in (levels 4-6) to visualize properties of functions and
their graphs, graphing calculators should be used to enhance their understanding and
skills rather than replace them.
7
Ma, Liping, Knowing and Teaching Elementary Mathematics, NYC: Taylor and Francis Routledge, 2010.
Milken, Lowell, A Matter of Quality: A Strategy for Answering the High Caliber of America’s Teachers, Santa
Monica, California: Milken Family Foundation, 1999.
9
Ma, p. 147.
10
National Center for Education Statistics, Pursuing Excellence: A Study of U.S. Fourth-Grade Mathematics and
Science Achievement in International Context. Accessed June 2000.
8
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Adult educators and adult learners should consider the available tools when presenting
or solving a problem. Adult learners should be familiar with tools appropriate for their
level to be able to make sound decisions about which of these tools would be helpful.
(See Standard for Mathematical Practice 5: Use appropriate tools strategically.)
Technology enables adult learners to communicate ideas within the classroom or to
search for information in external databases such as the Internet, an important
supplement to a program’s internal library resources. Technology can be especially
helpful in assisting students with special needs in all classrooms, at home, and in the
community.
Technology changes the mathematics to be learned, as well as when and how it is
learned. For example, currently available technology provides a dynamic approach to
such mathematical concepts as functions, rates of change, geometry, and averages that
was not possible in the past. Some mathematics becomes more important because
technology requires it, some becomes less important because technology replaces it,
and some becomes possible because technology allows it.
Guiding Principle 4: Equity
All students should have a high quality mathematics program that prepares them
for college and a career.
All Illinois adult education program students should have a high quality mathematics
program that meets the goals and expectations of these standards and addresses
students’ individual interests and talents. The standards provide clear signposts along
the way to the goal of college and career readiness for all students. The standards
provide for a broad range of students, from those requiring tutorial support to those with
talent in mathematics. To promote achievement of these standards, adult educators
should encourage classroom talk, reflection, use of multiple problem solving strategies,
and a positive disposition toward mathematics. They should have high expectations for
all students. At every level adult educators should act on the belief that every student
should learn challenging mathematics. Adult educators and student services, where
applicable, should advise adult learners about why it is important to take advanced
courses in mathematics and how this will prepare them for success in college and the
workplace.
All adult learners should have the benefit of quality instructional materials, good
libraries, and adequate technology. All students must have the opportunity to learn and
meet the same high standards. In order to meet the needs of the greatest range of
students, mathematics programs should provide the necessary intervention and support
for those students who are below or above specific level expectations. Practice and
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enrichment should extend beyond the classroom (i.e., tutoring sessions and
mathematics computerized activities that promote learning).
Because mathematics is the cornerstone of many disciplines, a comprehensive
curriculum should include applications to everyday life and modeling activities that
demonstrate the connections among disciplines. (See Standard for Mathematical
Practice 4: Model with mathematics.)
An important part of preparing students for college and careers is to ensure that they
have the necessary mathematics and problem-solving skills to make sound financial
decisions that they face in the world every day, including setting up a bank account;
understanding student loans; reading credit and debit statements; selecting the best buy
when shopping; and choosing the most cost effective cell phone plan based on monthly
usage.
Guiding Principle 5: Literacy Across the Content Areas
An effective mathematics program builds upon and develops students’ literacy
skills and knowledge.
Reading, writing, and communication skills are necessary elements of learning and
engaging in mathematics, as well as in other content areas. Supporting the
development of students’ literacy skills will allow them to deepen their understanding of
mathematics concepts and help them to determine the meanings of symbols, key terms,
and mathematics phrases, as well as to develop reasoning skills that apply across the
disciplines. In reading, adult educators should consistently support students’ ability to
gain and deepen understanding of concepts from written material by helping them
acquire comprehension skills and strategies, as well as specialized vocabulary and
symbols. Mathematics programs should make use of a variety of text materials and
formats, including textbooks, math journals, contextual math problems, and data
presented in a variety of media.
In writing, adult educators should consistently support students’ ability to reason and
achieve deeper understanding of concepts, and to express their understanding in a
focused, precise, and convincing manner. Mathematics classrooms should incorporate
a variety of written assignments ranging from math journals to formal written proofs. In
speaking and listening, adult educators should provide students with opportunities for
mathematical discourse using precise language to convey ideas, communicate
solutions, and support arguments. (See Standard for Mathematical Practice 6: Attend to
precision.)
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Guiding Principle 6: Assessment
Effective assessment of student learning in mathematics should take many forms
to inform instruction and learning.
A comprehensive assessment program is an integral component of an instructional
program. It provides adult learners with frequent feedback on their performance, adult
educators with diagnostic tools for gauging students’ depth of understanding of
mathematical concepts and skills, and administrators with a means for measuring adult
learners’ achievement.
Assessments take a variety of forms, require varying amounts of time, and address
different aspects of student learning. Having students “think aloud” or talk through their
solutions to problems permits identification of gaps in knowledge and errors in
reasoning. By observing adult learners as they work, adult educators can gain insight
into students’ abilities to apply appropriate mathematical concepts and skills, make
conjectures, and draw conclusions. Homework, mathematics journals, portfolios, oral
performances, and group projects offer additional means for capturing students’
thinking, knowledge of mathematics, facility with the language of mathematics, and
ability to communicate what they know to others. Tests and quizzes assess knowledge
of mathematical facts, operations, concepts, and skills, and their efficient application to
problem solving; they can also pinpoint areas in need of more practice or teaching.
Taken together, the results of these different forms of assessment provide rich profiles
of adult learners’ achievements in mathematics and serve as the basis for identifying
curricula and instructional approaches to best develop their talents.
Assessment should also be a major component of the learning process. As students
help identify goals for lessons or investigations, they gain greater awareness of what
they need to learn and how they will demonstrate that learning. Engaging adult learners
in this kind of goal-setting can help them reflect on their own work, understand the
standards to which they are held accountable, and take ownership of their learning.
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How to Read the Six NRS Level Standards11
Domains are larger groups of related standards. Standards from different domains may
sometimes be closely related.
Clusters summarize groups of related standards. Note that standards from different
clusters may sometimes be closely related, because mathematics is a connected
subject.
Standards define what students should understand and be able to do.
Domain
Counting and Cardinality / Numeracy (CC)
Compare numbers.
1.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to
the number of objects in another group (e.g., by using matching and counting strategies).
Include groups with up to ten objects.
Cluster
Standard
1.CC.7 Compare two numbers between 1 and 10 presented as written numerals.
The number of each standard is listed in the left hand column of each NRS level. The
first number refers to the NRS level of the standard. The next two letters indicate the
standard area by code; and the final number indicates the number of the skill within
each domain. For example, “1.CC.6” labels NRS Level 1 Math standard #6 in the
Counting and Cardinality / Numeracy section.
These standards do not dictate curriculum or teaching methods. For example, just
because topic A appears before topic B in the standards for a given level, it does not
necessarily mean that topic A must be taught before topic B. An adult educator might
prefer to teach topic B before topic A, or might choose to highlight connections by
teaching topic A and topic B at the same time. Or, an adult educator might prefer to
teach a topic of his or her own choosing that leads, as a byproduct, to students reaching
the standards for topics A and B.
What adult students can learn at any particular level depends upon what they have
learned before. Ideally then, each standard in this document might have been phrased
in the form, “Students who already know A should next come to learn B.” But at present
this approach is unrealistic—not least because existing education research cannot
specify all such learning pathways. Of necessity, therefore, level placements for specific
topics have been made on the basis of state and international comparisons and the
11
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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collective experience and collective professional judgment of educators, researchers,
and mathematicians. One promise of the Illinois ABE/ASE state standards is that over
time they will allow research on learning progressions to inform and improve the design
of standards to a much greater extent than is possible today. Learning opportunities will
continue to vary across the state of Illinois adult education programs, and educators
should make every effort to meet the needs of all students based on their current
understanding.
The reference column lists the standard to which the ABE/ASE standard is aligned.
Reference sources include: Common Core State Standard (CC)12, States’ Career
Cluster Initiative Essential Knowledge and Skill Statements (ESS)13, and the
International Society for Technology in Education’s National Educational Technology
Standards for Students (NETS•S)14 and the College and Career Readiness (CCR)
Standards for Adult Education15.
12
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
13
Career Cluster™ Knowledge and Skills, 2008, www.careertech.org.
14
National Educational Technology Standards for Students, Second Edition, ©2007, ISTE® (International Society for Technology in Education),
www.iste.org.
15
College and Career Readiness (CCR) for Adult Education, 2013, www.lincs.ed.gov/publications/pdf/CCRStandardsAdultEd.pdf.
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The Standards for Mathematical Content
NRS Levels 1- 6
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NRS Level 1 Overview
Counting and Cardinality / Numeracy
(CC)
 Know number names and the count
sequence.
 Count to tell the number of objects
 Compare numbers


Tell and write time
Represent and interpret data
Geometry (G)
 Identify and describe shapes
 Analyze, compare, create, and
Operations and Algebraic Thinking (OA)
compose shapes
 Understand addition as putting
 Reason with shapes and their
together and adding to, and
attributes
understand subtraction as taking
apart and taking from
 Represent and solve problems
involving addition and subtraction
Mathematical Practices
 Understand and apply properties
and the relationship between
1. Make sense of problems and
addition and subtraction
persevere in solving them.
 Add and subtract within 20
 Work with addition and subtraction
2. Reason abstractly and quantitatively.
equations
Number and Operations in Base Ten
(NBT)
 Extend the counting sequence
 Work with numbers 11-19 and tens
to gain foundations for place value
 Use place value understanding and
properties of operations to add and
subtract
Measurement and Data (MD)
 Describe and compare measurable
attributes
 Classify objects and count the
number of objects in each category
 Measure lengths indirectly and by
iterating length units
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
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OVERVIEW EXPLANATION OF MATHEMATICS
NRS Level 1 – Beginning ABE Literacy
(Grade Levels 0 – 1.9)
At the Beginning ABE Literacy Level, instructional time should focus on four critical
areas16:
1. Developing understanding of addition, subtraction, and strategies for addition and
subtraction within 20;
2. Developing understanding of whole number relationships and place value,
including grouping in tens and ones;
3. Developing understanding of linear measurement and measuring lengths as
iterating length units; and
4. Describing, reasoning about attributes of, composing and decomposing
geometric shapes.
(1) Adult learners develop strategies for adding and subtracting whole numbers. They
use a variety of models, including discrete objects and length-based models (e.g.,
cubes connected to form lengths), to model add-to, take-from, put-together, take-apart,
and compare situations to develop meaning for the operations of addition and
subtraction, and to develop strategies to solve arithmetic problems with these
operations. Students understand connections between counting and addition and
subtraction (e.g., adding two is the same as counting on two). They use properties of
addition to add whole numbers and to create and use increasingly sophisticated
strategies based on these properties (e.g., “making tens”) to solve addition and
subtraction problems within 20. By comparing a variety of solution strategies, students
build their understanding of the relationship between addition and subtraction.
(2) Adult learners develop, discuss, and use efficient, accurate, and generalizable
methods to add within 100 and subtract multiples of 10. They compare whole numbers
(at least to 100) to develop understanding of and solve problems involving their relative
sizes. They think of whole numbers between 10 and 100 in terms of tens and ones
(especially recognizing the numbers 11 to 19 as composed of a ten and some ones).
Through activities that build number sense, they understand the order of the counting
numbers and their relative magnitudes.
(3) Adult learners develop an understanding of the meaning and processes of
measurement, including underlying concepts such as iterating (the mental activity of
building up the length of an object with equal-sized units) and the transitivity principle for
indirect measurement.
16
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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(4) Adult learners compose and decompose plane or solid figures (e.g., put two
triangles together to make a quadrilateral) and build understanding of part-whole
relationships as well as the properties of the original and composite shapes. As they
combine shapes, they recognize them from different perspectives and orientations,
describe their geometric attributes, and determine how they are alike and different, to
develop the background for measurement and for initial understandings of properties
such as congruence and symmetry.
The Standards for Mathematical Practice complement the content standards so that
students increasingly engage with the subject matter as they grow in mathematical
understanding and expertise.
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NRS Level 1 – Beginning ABE Literacy
(Grade Levels 0-1.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This Skill
COUNTING AND CARDINALITY / NUMERACY (CC)
Know number names and the count sequence.
1.CC.1
Count to 100 by ones and by tens.
1.CC.2
CC.K.CC.1
ESS01.03.01
CC.K.CC.2
ESS01.03.01
Count forward beginning from a given
number within the known sequence
(instead of having to begin at 1).
1.CC.3
Write numbers from 0 to 20. Represent a CC.K.CC.3
number of objects with a written numeral
ESS01.03.01
0-20 (with 0 representing a count of no
objects).
Count to tell the number of objects.
1.CC.4
Understand the relationship between
CC.K.CC.4
numbers and quantities; connect counting
to cardinality.
a. When counting objects, say the
number names in the standard order,
pairing each object with one and only
one number name and each number
name with one and only one object.
b. Understand that the last number name
said tells the number of objects
counted. The number of objects is the
same regardless of their arrangement
or the order in which they were
counted.
c. Understand that each successive
number name refers to a quantity that
is one larger.
1.CC.5
Count to answer “how many?” questions
about as many as 20 things arranged in a
line, a rectangular array, or a circle, or as
many as 10 things in a scattered
configuration; given a number from 1–20,
count out that many objects.
CC.K.CC.5
ESS01.03.01
Counting dimes, $10
bills
Counting page
numbers read
Counting items in a
group
Counting coins to
pay for a purchase
Telling which
address falls in a
given block, knowing
the first number on
the block
Counting the number
of loose coins in a
pile
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Compare numbers.
1.CC.6
Identify whether the number of objects in
one group is greater than, less than, or
equal to the number of objects in another
group (e.g., by using matching and
counting strategies). Include groups with
up to ten objects.
1.CC.7
Compare two numbers between 1 and 10
presented as written numerals.
CC.K.CC.6
ESS01.03.03
Separating loose
coins into like piles
and counting the
number in each
CC.K.CC.7
ESS01.03.03
Comparing price
tags on two items
OPERATIONS AND ALGEBRAIC THINKING (OA)
Understand addition as putting together and adding to, and understand subtraction as
taking apart and taking from.
1.OA.1
Represent addition and subtraction with
CC.K.OA.1
Figuring the number
objects, fingers, mental images,
ESS01.03.02
of hours of work or
drawings, sounds (e.g., claps), and acting
sleep by using
out situations, verbal explanations,
fingers to count
expressions, or equations. Drawings
need not show details, but should show
the mathematics in the problem.
1.OA.2
Solve addition and subtraction word
CC.K.OA.2
Counting money and
problems, and add and subtract within 10 ESS01.03.02
making change
(e.g., by using objects or drawings to
represent the problem).
1.OA.3
Decompose numbers less than or equal
CC.K.OA.3
Using manipulatives
to 10 into pairs in more than one way
ESS01.03.02
to establish number
(e.g., by using objects or drawings), and
relationships
record each decomposition by a drawing
or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
1.OA.4
For any number from 1 to 9, find the
CC.K.OA.4
Using manipulatives
number that makes 10 when added to the ESS01.03.02
to establish number
given number (e.g., by using objects or
relationships
drawings), and record the answer with a
drawing or equation.
1.OA.5
Fluently add and subtract within 5.
CC.K.OA.5
Counting money and
ESS01.03.02
making change
Represent and solve problems involving addition and subtraction.
1.OA.6
Use addition and subtraction within 20 to CC.1.OA.1
Working out the
solve word problems involving situations
ESS01.03.02
shortfall in numbers
of adding to, taking from, putting together,
(e.g., eggs for a
taking apart, and comparing, with
recipe, plants to fill a
unknowns in all positions (e.g., by using
display tray, cups to
objects, drawings, and equations with a
serve visitors)
symbol for the unknown number to
represent the problem).
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1.OA.7
Solve word problems that call for addition CC.1.OA.2
Finding the total
of three whole numbers whose sum is
ESS01.03.02
price of 3 items
less than or equal to 20 (e.g., by using
CCR.OA.A
ordered from a menu
objects, drawings, and equations with a
symbol for the unknown number to
represent the problem).
Understand and apply properties and the relationship between addition and
subtraction.
1.OA.8
Apply properties of operations as
CC.1.OA.3
Placing same
strategies to add and subtract.
ESS01.03.02
number of cookies
Examples: If 8 + 3 = 11 is known, then
CCR.OA.A
on different shaped
3 + 8 = 11 is also known (commutative
trays
property of addition). To add 2 + 6 + 4,
the second two numbers can be added to
make a ten, so 2 + 6 + 4 = 2 + 10 = 12
(associative property of addition).
Students need not use formal terms for
these properties.
1.OA.9
Understand subtraction as an unknownCC.1.OA.4
Figuring change to
addend problem. For example, subtract
ESS01.03.02
receive from a $10
10 – 8 by finding the number that makes
CCR.OA.A
bill
10 when added to 8.
Add and subtract within 20.
1.OA.10
Relate counting to addition and
CC.1.OA.5
Watching a digital
subtraction (e.g., by counting on 2 to add ESS01.03.02
clock count down the
2).
CCR.OA.A
time
1.OA.11
Add and subtract within 20,
CC.1.OA.6
Paying a $12 charge
demonstrating fluency for addition and
ESS01.03.02
with a $10 bill and
subtraction within 10. Use strategies such CCR.OA.A
two $1 dollar bills
as counting on; making ten (e.g., 8 + 6 =
8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13 – 4 = 13
– 3 – 1 = 10 – 1 = 9); using the
relationship between addition and
subtraction (e.g., knowing that 8 + 4 = 12,
one knows 12 – 8 = 4); and reading
equivalent but easier or known sums
(e.g., adding 6 + 7 by creating the known
equivalent 6 + 6 + 1 = 12 + 1 = 13).
Work with addition and subtraction equations.
1.OA.12
Understand the meaning of the equal
CC.1.OA.7
Counting money and
sign, and determine if equations involving ESS01.03.02
making change
addition and subtraction are true or false. CCR.OA.A
For example, which of the following
equations are true and which are false? 6
= 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
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1.OA.13
Determine the unknown whole number in
an addition or subtraction equation
relating three whole numbers. For
example, determine the unknown number
that makes the equation true in each of
the equations 8 + ? = 11, 5 = ? – 3,
6 + 6 = ?.
CC.1.OA.8
ESS01.03.03
CCR.OA.A
Test taking when
seeking employment
Helping children with
homework
NUMBER AND OPERATIONS IN BASE TEN (NBT)
Extend the counting sequence.
1.NBT.1
Count to 120, starting at any number less CC.1.NBT.1
Counting page
than 120. In this range, read and write
ESS01.03.01
numbers read at one
numerals and represent a number of
time, starting from
objects with a written numeral.
first page read
Work with numbers 11-19 and tens to gain a foundation and understand place value.
1.NBT.2
Understand that the two digits of a twoCC.K.NBT.1
Using mental math to
digit number represent amounts of tens
CC.1.NBT.2
check that correct
and ones. Understand the following as
ESS01.03.01
change was received
special cases:
CCR.NBT.A
a. 10 can be thought of as a bundle
of ten ones—called a “ten.”
b. The numbers from 11 to 19 are
composed of a ten and one, two,
three, four, five, six, seven, eight,
or nine ones.
c. The numbers 10, 20, 30, 40, 50,
60, 70, 80, 90 refer to one, two,
three, four, five, six, seven, eight,
or nine tens (and 0 ones).
1.NBT.3
Compare two two-digit numbers based on CC.1.NBT.3
meanings of the tens and ones digits,
ESS01.03.03
recording the results of comparisons with CCR.NBT.A
the symbols >, =, and <.
Telling which
address falls in a
block, knowing the
house number
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Use place value understanding and properties of operations to add and subtract.
1.NBT.4
Add within 100, including adding a twoCC.1.NBT.4
Calculating the
digit number and a one-digit number, and ESS01.03.02
production shortfall
adding a two-digit number and a multiple CCR.NBT.A
from a daily target
of 10, using concrete models or drawings
and strategies based on place value,
Performing mental
properties of operations, and/or the
addition
relationship between addition and
subtraction; relate the strategy to a
Verifying deposits in
written method and explain the reasoning
a checking account
used. Understand that in adding two-digit
numbers, one adds tens and tens, ones
and ones; and sometimes it is necessary
to compose a ten.
1.NBT.5
Given a two-digit number, mentally find
CC.1.NBT.5
Adding using mental
10 more or 10 less than the number,
ESS01.03.02
math
without having to count; explain the
CCR.NBT.A
reasoning used.
1.NBT.6
Subtract multiples of 10 in the range 10–
CC.1.NBT.6
Counting money and
90 from multiples of 10 in the range 10–
ESS01.03.02
making change
90 (positive or zero differences), using
CCR.NBT.A
concrete models or drawings and
Changing radio
strategies based on place value,
stations
properties of operations, and/or the
relationship between addition and
subtraction; relate the strategy to a
written method and explain the reasoning
used.
MEASUREMENT AND DATA (MD)
Describe and compare measurable attributes.
1.MD.1
Describe measurable attributes of
objects, such as length or weight.
Describe several measurable attributes of
a single object.
1.MD.2
Directly compare two objects with a
measurable attribute in common, to see
which object has “more of”/“less of” the
attribute, and describe the difference. For
example, directly compare the heights of
two people and describe one person as
taller/shorter.
CC.K.MD.1
Describing a
rectangular photo or
frame
CC.K.MD.2
ESS01.03.03
ESS01.03.04
Describing seasons,
daylight savings
time, or tides
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Classify objects and count the number of objects in each category.
1.MD.3
Classify objects into given categories;
CC.K.MD.3
count the numbers of objects in each
ESS01.03.01
category and sort the categories by
count.
Measure lengths indirectly and by iterating length units.
1.MD.4
Order three objects by length; compare
CC.1.MD.1
the lengths of two objects indirectly by
using a third object.
1.MD.5
Express the length of an object as a
CC.1.MD.2
whole number of length units, by laying
ESS01.03.04
multiple copies of a shorter object (the
CCR.MD.A
length unit) end to end; understand that
the length measurement of an object is
the number of same-size length units that
span it with no gaps or overlaps. Limit to
contexts where the object being
measured is spanned by a whole number
of length units with no gaps or overlaps.
Tell and write time.
1.MD.6
Tell and write time in hours and halfCC.1.MD.3
hours using analog and digital clocks.
Represent and interpret data.
1.MD.7
Organize, represent, and interpret data
with up to three categories; ask and
answer questions about the total
number of data points, how many in
each category, and how many more or
less are in one category than in
another.
CC.1.MD.4
ESS01.03.06
CCR.MD.A
Sorting laundry or
bottles for the
recycling facility
Understanding a
child’s growth chart
Measuring the length
and width of a photo
Reading a bus
schedule that uses
a.m. and p.m.
Understanding a
child’s growth chart
GEOMETRY (G)
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes,
cones, cylinders, and spheres).
1.G.1
Describe objects in the environment using CC.K.G.1
Recognizing the
names of shapes, and describe the
shape and meaning
relative positions of these objects using
of a triangular yield
terms such as above, below, beside, in
sign and other
front of, behind, and next to.
shapes in buildings
and everyday
structures
1.G.2
Correctly name shapes regardless of their CC.K.G.2
Identifying things by
orientations or overall size.
shape
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1.G.3
Identify shapes as two-dimensional (lying CC.K.G.3
in a plane, “flat”) or three- dimensional
(“solid”).
Analyze, compare, create, and compose shapes.
1.G.4
Analyze and compare two- and threeCC.K.G.4
dimensional shapes, in different sizes and CCR.G.A
orientations, using informal language to
describe their similarities, differences,
parts (e.g., number of sides and
vertices/“corners”) and other attributes
(e.g., having sides of equal length).
Building a three
dimensional model
from flat material
1.G.5
Using shapes to
replicate campus
map and roadways
Building furniture
Model shapes in the world by building
shapes from components and drawing
shapes.
1.G.6
Compose simple shapes to form larger
shapes. For example, “Can you join these
two triangles with full sides touching to
make a rectangle?”
Reason with shapes and their attributes.
1.G.7
Distinguish between defining attributes
(e.g., triangles are closed and threesided) versus non-defining attributes
(e.g., color, orientation, overall size); build
and draw shapes to possess defining
attributes.
1.G.8
Compose two-dimensional shapes
(rectangles, squares, trapezoids,
triangles, half-circles, and quarter-circles)
or three-dimensional shapes (cubes, right
rectangular prisms, right circular cones,
and right circular cylinders) to create a
composite shape, and compose new
shapes from the composite shape.
Students do not need to learn formal
names such as “right rectangular prism.”
1.G.9
Partition circles and rectangles into two
and four equal shares, describe the
shares using the words halves, fourths,
and quarters, and use the phrases half of,
fourth of, and quarter of. Describe the
whole as two of or four of the shares.
Understand for these examples that
decomposing into more equal shares
creates smaller shares.
CC.K.G.5
CC.K.G.6
Using a “Rules of the
Road” book to
describe shapes of
road signs
CC.1.G.1
Checking for quality
control
CC.1.G.2
CCR.G.A
Building furniture
CC.1.G.3
Cutting pizza, cake,
and brownies
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NRS Level 2 Overview
Operations and Algebraic Thinking (OA)
 Represent and solve problems involving addition
and subtraction
 Add and subtract within 20
 Work with equal groups of objects to gain
foundations for multiplication
 Represent and solve problems involving
multiplication and division
 Understand properties of multiplication and the
relationship between multiplication and division
 Multiply and divide within 100
 Solve problems involving the four operations and
identify and explain patterns in arithmetic
Number and Operations in Base Ten (NBT)
 Understand place value
 Use place value understanding and properties of
operations to add and subtract and to perform
multi-digit arithmetic
Number and Operations – Fractions (NF)
 Develop understanding of fractions as numbers
Mathematical Practices
1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express regularity in
repeated reasoning.
Measurement and Data (MD)
 Measure and estimate lengths in standard units
 Relate addition and subtraction to length
 Work with time and money
 Represent and interpret data
 Solve problems involving measurement and estimation of intervals of time, liquid
volumes, and masses of objects
 Geometric measurement: understand concepts of area and relate area to
multiplication and to addition
 Geometric measurement: recognize perimeter as an attribute of plan figures an
distinguish between linear and area measures
Geometry (G)
 Reason with shapes and their attributes
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OVERVIEW EXPLANATION OF MATHEMATICS
NRS Level 2 – Beginning Basic Education
(Grade Levels 2.0 – 3.9)
At the Beginning Basic Education Level, instructional time should focus on six critical
areas17:
1. Extending understanding of base-ten notation;
2. Building fluency with addition and subtraction;
3. Developing understanding of multiplication and division and strategies for
multiplication and division within 100;
4. Developing understanding of fractions, especially unit fractions (fractions with
numerator 1);
5. Using standard units of measure and developing understanding of the structure
of rectangular arrays and of area; and
6. Describing and analyzing one- and two-dimensional shapes.
(1) Adult learners extend their understanding of the base-ten system. This includes
ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as
number relationships involving these units, including comparing. Students understand
multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits
in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8
hundreds + 5 tens + 3 ones).
(2) Adult learners use their understanding of addition to develop fluency with addition
and subtraction within 100. They solve problems within 1000 by applying their
understanding of models for addition and subtraction, and they develop, discuss, and
use efficient, accurate, and generalizable methods to compute sums and differences of
whole numbers in base-ten notation, using their understanding of place value and the
properties of operations. They select and accurately apply methods that are appropriate
for the context and the numbers involved to mentally calculate sums and differences for
numbers with only tens or only hundreds.
(3) Adult learners develop an understanding of the meanings of multiplication and
division of whole numbers through activities and problems involving equal-sized groups,
arrays, and area models; multiplication is finding an unknown product, and division is
finding an unknown factor in these situations. For equal-sized group situations, division
can require finding the unknown number of groups or the unknown group size. Students
use properties of operations to calculate products of whole numbers, using increasingly
17
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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Illinois Community College Board, 5/2014
sophisticated strategies based on these properties to solve multiplication and division
problems involving single-digit factors. By comparing a variety of solution strategies,
students learn the relationship between multiplication and division.
(4) Adult learners develop an understanding of fractions, beginning with unit fractions.
Students view fractions in general as being built out of unit fractions, and they use
fractions along with visual fraction models to represent parts of a whole. Students
understand that the size of a fractional part is relative to the size of the whole. For
example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a
larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when
the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is
divided into 5 equal parts. Students are able to use fractions to represent numbers
equal to, less than, and greater than one. They solve problems that involve comparing
fractions by using visual fraction models and strategies based on noticing equal
numerators or denominators.
(5) Adult learners recognize the need for standard units of measure (centimeter and
inch), and they use rulers and other measurement tools with the understanding that
linear measure involves an iteration of units. When building, students recognize area as
an attribute of two-dimensional regions. They measure the area of a shape by finding
the total number of same-size units of area required to cover the shape without gaps or
overlaps, a square with sides of unit length being the standard unit for measuring area.
Students understand that rectangular arrays can be decomposed into identical rows or
into identical columns. By decomposing rectangles into rectangular arrays of squares,
students connect area to multiplication, and justify using multiplication to determine the
area of a rectangle.
(6) Adult learners describe, analyze, and compare properties of one- and twodimensional shapes. They compare and classify shapes by their sides and angles, and
connect these with definitions of shapes. They investigate, describe, and reason about
decomposing and combining shapes to make other shapes. Through building, drawing,
and analyzing two- and three-dimensional shapes, students develop a foundation for
understanding area, volume, congruence, similarity, and symmetry. Students also
relate their fraction work to geometry by expressing the area of part of a shape as a unit
fraction of the whole.
The Standards for Mathematical Practice complement the content standards so that
students increasingly engage with the subject matter as they grow in mathematical
understanding and expertise.
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NRS Level 2 – Beginning Basic Education
(Grade Levels 2.0-3.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This Skill
OPERATIONS AND ALGEBRAIC THINKING (OA)
Represent and solve problems involving addition and subtraction.
2.OA.1
Use addition and subtraction within 100 to CC.2.OA.1
Carrying out a stock
solve one- and two-step word problems
ESS01.03.02
inventory
involving situations of adding to, taking
ESS01.03.05
from, putting together, taking apart, and
CCR.OA.B
Checking grocery
comparing, with unknowns in all positions
receipt against
(e.g., by using drawings and equations
purchases
with a symbol for the unknown number to
represent the problem).
Add and subtract within 20.
2.OA.2
Fluently add and subtract within 20 using CC.2.OA.2
Estimating the bill at a
mental strategies. Know from memory all ESS01.03.02
restaurant
sums of two one-digit numbers.
CCR.OA.B
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3
Determine whether a group of objects (up CC.2.OA.3
Telling which side of
to 20) has an odd or even number of
ESS01.03.02
a street a house will
members (e.g., by pairing objects or
ESS01.03.03
be on from its number
counting them by 2s); write an equation to
express an even number as a sum of two
equal addends.
2.OA.4
Use addition to find the total number of
CC.2.OA.4
Finding out how many
objects arranged in rectangular arrays
ESS01.03.02
chairs are needed for
with up to 5 rows and up to 5 columns;
a meeting
write an equation to express the total as a
sum of equal addends.
Represent and solve problems involving multiplication and division.
2.OA.5
Interpret products of whole numbers (e.g., CC.3.OA.1
Determining how
interpret 5 x 7 as the total number of
ESS01.03.02
many pieces of pie
objects in 5 groups of 7 objects each).
CCR.OA.B
you will have with
For example, describe a context in which
multiple pies
a total number of objects can be
expressed as 5 x 7.
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2.OA.6
Interpret whole-number quotients of
CC.3.OA.2
Splitting a restaurant
whole numbers (e.g., interpret 56 ÷ 8 as
ESS01.03.02
bill (check) into equal
the number of objects in each share when CCR.OA.B
parts for 2, 3, 4, 5 or
56 objects are partitioned equally into 8
more people
shares, or as a number of shares when
56 objects are partitioned into equal
shares of 8 objects each). For example,
describe a context in which a number of
shares or a number of groups can be
expressed as 56 ÷ 8.
2.OA.7
Use multiplication and division within 100 CC.3.OA.3
Determining the total
to solve word problems in situations
ESS01.03.02
amount of money
involving equal groups, arrays, and
ESS01.03.05
when each person
measurement quantities (e.g., by using
CCR.OA.B
pays an equal fee
drawings and equations with a symbol for
the unknown number to represent the
problem).
2.OA.8
Determine the unknown whole number in CC.3.OA.4
Recognizing
a multiplication or division equation
ESS01.03.05
multiplication as
relating three whole numbers. For
CCR.OA.B
repeated addition
example, determine the unknown number
that makes the equation true in each of
the equations 8 × ? = 48, 5 = ? = 3, 6 × 6
=?
Understand properties of multiplication and the relationship between multiplication and
division.
2.OA.9
Apply properties of operations as
CC.3.OA.5
Calculating total
strategies to multiply and divide.
ESS01.03.02
number (e.g., three
Examples: If 6 × 4 = 24 is known, then 4
CCR.OA.B
days a week for four
× 6 = 24 is also known (commutative
weeks)
property of multiplication). 3 × 5 × 2 can
be found by 3 × 5 = 15, then 15 × 2 = 30,
or by 5 × 2 = 10, then 3 × 10 = 30
(associative property of multiplication).
Knowing that 8 × 5 = 40 and 8 × 2 = 16,
one can find 8 × 7 as 8 × (5 + 2) = (8 × 5)
+ (8 × 2) = 40 + 16 = 56 (distributive
property). Students need not use formal
terms for these properties.
2.OA.10
Understand division as an unknownCC.3.OA.6
Working out how
factor problem. For example, find 32 ÷ 8
ESS01.03.02
many cars are
by finding the number that makes 32
CCR.OA.B
needed to transport a
when multiplied by 8.
group of people
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Multiply and divide within 100.
2.OA.11
Fluently multiply and divide within 100,
CC.3.OA.7
Dividing work time for
using strategies such as the relationship
ESS01.03.02
employees
between multiplication and division (e.g.,
CCR.OA.B
knowing that 8 ×5 = 40, one knows 40 ÷ 5
= 8) or properties of operations. Know
from memory all products of two one-digit
numbers.
Solve problems involving the four operations, and identify and explain patterns in
arithmetic.
2.OA.12
Solve two-step word problems using the
CC.3.OA.8
Estimating amount of
four operations. Represent these
ESS01.03.02
purchase to nearest
problems using equations with a letter
ESS01.03.05
10 dollars
standing for the unknown quantity.
CCR.OA.B
Assess the reasonableness of answers
Estimating distances
using mental computation and estimation
between cities
strategies including rounding. This
standard is limited to problems posed
Giving ballpark
with whole numbers and having whole
figures for numbers in
number answers; students should know
a crowd
how to perform operations in the
conventional order when there are no
parentheses to specify a particular order
(order of operations).
2.OA.13
Identify arithmetic patterns (including
CC.3.OA.9
Laying tile on a floor
patterns in the addition table or
ESS01.03.02
multiplication table), and explain them
ESS01.03.03
using properties of operations. For
CCR.OA.B
example, observe that 4 times a number
is always even, and explain why 4 times a
number can be decomposed into two
equal addends.
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NUMBER AND OPERATIONS IN BASE TEN (NBT)
Understand place value.
2.NBT.1
Understand that the three digits of a
three-digit number represent amounts of
hundreds, tens, and ones (e.g., 706
equals 7 hundreds, 0 tens, and 6 ones).
Understand the following as special
cases:
a. 100 can be thought of as a bundle
of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400,
500, 600, 700, 800, 900 refer to
one, two, three, four, five, six,
seven, eight, or nine hundreds
(and 0 tens and 0 ones).
2.NBT.2
Count within 1000; skip-count by 5s, 10s,
and 100s.
2.NBT.3
CC.2.NBT.1
ESS01.03.01
CCR.NBT.B
Exchanging money
into small bills or
coins
CC.2.NBT.2
ESS01.03.01
CCR.NBT.B
CC.2.NBT.3
ESS01.03.01
CCR.NBT.B
CC.2.NBT.4
ESS01.03.03
CCR.NBT.B
Reading graphs
accurately
Read and write numbers to 1000 using
Writing a rent or
base-ten numerals, number names, and
tuition check
expanded form.
2.NBT.4
Compare two three-digit numbers based
Balancing a
on meanings of the hundreds, tens, and
checkbook
ones digits, using >, =, and < symbols to
record the results of comparisons.
Use place value understanding and properties of operations to add and subtract and to
perform multi-digit arithmetic.
2.NBT.5
Add up to four two-digit numbers using
CC.2.NBT.6
Balancing a
strategies based on place value and
ESS01.03.02
checkbook
CCR.NBT.B
properties of operations.
2.NBT.6
Add and subtract within 1000, using
CC.2.NBT.7
Balancing a
concrete models or drawings and
ESS01.03.02
checkbook
strategies based on place value,
CCR.NBT.B
properties of operations, and/or the
Finding how much
relationship between addition and
money is left after
subtraction; relate the strategy to a
shopping at different
written method. Understand that in adding
places
or subtracting three-digit numbers, one
adds or subtracts hundreds and
hundreds, tens and tens, ones and ones;
and sometimes it is necessary to
compose or decompose tens or
hundreds.
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2.NBT.7
Mentally add 10 or 100 to a given number
100–900, and mentally subtract 10 or 100
from a given number 100–900.
CC.2.NBT.8
ESS01.03.02
CCR.NBT.B
2.NBT.8
Explain why addition and subtraction
strategies work, using place value and
the properties of operations.
(Explanations may be supported by
drawings or objects.)
Use place value understanding to round
whole numbers to the nearest 10 or 100.
Fluently add and subtract within 1000
using strategies and algorithms based on
place value, properties of operations,
and/or the relationship between addition
and subtraction.
Multiply one-digit whole numbers by
multiples of 10 in the range 10–90 (e.g., 9
× 80, 5 × 60) using strategies based on
place value and properties of operations.
CC.2.NBT.9
ESS01.03.02
CCR.NBT.B
2.NBT.9
2.NBT.10
2.NBT.11
Determining the new
temperature for
baking with recipe
changes
CC.3.NBT.1
CCR.NBT.B
CC.2.NBT.5
CC.3.NBT.2
ESS01.03.02
CCR.NBT.B
CC.3.NBT.3
ESS01.03.02
CCR.NBT.B
NUMBERS AND OPERATIONS IN FRACTIONS (NF)
Develop understanding of fractions as numbers.
Understand a fraction 1/b as the quantity
2.NF.1
formed by 1 part when a whole is
partitioned into b equal parts; understand
a fraction a/b as the quantity formed by a
parts of size 1/b.
CC.3.NF.1
ESS01.03.01
CCR.NF.B
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2.NF.2
2.NF.3
Understand a fraction as a number on the
number line; represent fractions on a
number line diagram.
a. Represent a fraction 1/b on a
number line diagram by defining
the interval from 0 to 1 as the
whole and partitioning it
into b equal parts. Recognize that
each part has size 1/b and that the
endpoint of the part based at 0
locates the number 1/b on the
number line.
b. Represent a fraction a/b on a
number line diagram by marking
off a lengths 1/b from 0. Recognize
that the resulting interval has
size a/b and that its endpoint
locates the number a/b on the
number line.
Explain equivalence of fractions in special
cases, and compare fractions by
reasoning about their size.
a. Understand two fractions as
equivalent (equal) if they are the
same size or the same point on a
number line.
b. Recognize and generate simple
equivalent fractions (e.g., 1/2 =
2/4, 4/6 = 2/3). Explain why the
fractions are equivalent (e.g., by
using a visual fraction model).
c. Express whole numbers as
fractions, and recognize fractions
that are equivalent to whole
numbers. Examples: Express 3 in
the form 3 = 3/1; recognize that 6/1
= 6; locate 4/4 and 1 at the same
point of a number line diagram.
d. Compare two fractions with the
same numerator or the same
denominator by reasoning about
their size. Recognize that
comparisons are valid only when
the two fractions refer to the same
whole.
CC.3.NF.2
ESS01.03.01
CCR.NF.B
Following a recipe
CC.3.NF.3
ESS01.03.03
CCR.NF.B
Recording the results
of comparisons with
the symbols >, =, or
<, and justify the
conclusions (e.g., by
using a visual fraction
model)
Using a ¼ cup
measure to add ¾
cup of flour to a
recipe
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MEASUREMENT AND DATA (MD)
Measure and estimate lengths in standard units.
2.MD.1
Measure the length of an object by
selecting and using appropriate tools
such as rulers, yardsticks, meter sticks,
and measuring tapes.
2.MD.2
Measure the length of an object twice,
using length units of different lengths for
the two measurements; describe how the
two measurements relate to the size of
the unit chosen.
2.MD.3
Estimate lengths using units of inches,
feet, centimeters, and meters.
CC.2.MD.1
ESS01.03.04
Measuring a room for
new carpet
CC.2.MD.2
ESS01.03.04
CCR.MD.B
Sorting by size
CC.2.MD.3
ESS01.03.04
CCR.MD.B
2.MD.4
Measure to determine how much longer
CC.2.MD.4
one object is than another, expressing the ESS01.03.03
length difference in terms of a standard
CCR.MD.B
length unit.
Relate addition and subtraction to length.
2.MD.5
Use addition and subtraction within 100 to CC.2.MD.5
solve word problems involving lengths
ESS01.03.02
that are given in the same units (e.g., by
ESS01.03.04
using drawings (such as drawings of
rulers) and equations with a symbol for
the unknown number to represent the
problem).
2.MD.6
Represent whole numbers as lengths
CC.2.MD.6
from 0 on a number line diagram with
ESS01.03.01
equally spaced points corresponding to
CCR.MD.B
the numbers 0, 1, 2... and represent
whole-number sums and differences
within 100 on a number line diagram.
Work with time and money.
2.MD.7
Tell and write time from analog and digital CC.2.MD.7
clocks to the nearest five minutes, using
a.m. and p.m.
2.MD.8
Solve word problems involving dollar bills, CC.2.MD.8
quarters, dimes, nickels, and pennies,
ESS01.03.02
using $ and ¢ symbols appropriately.
Example: If you have 2 dimes and 3
pennies, how many cents do you have?
Hanging a picture or
an award on the wall
Deciding whether a
sheet of wall paper is
long enough for a wall
Model drawings
Computing hours
worked or pay for
babysitter
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Illinois Community College Board, 5/2014
Represent and interpret data.
2.MD.9
Draw a scaled picture graph and a scaled CC.2.MD.10
bar graph to represent a data set with
CC.3.MD.3
several categories. Solve one- and twoESS01.03.06
step “how many more” and “how many
CCR.MD.B
less” problems using information
presented in scaled bar graphs. For
example, draw a bar graph in which each
square in the bar graph might represent 5
pets.
2.MD.10
Generate measurement data by
CC.2.MD.9
Dividing a dollar bill in
measuring lengths of several objects
CC.3.MD.4
halves and quarters,
using rulers marked with halves and
ESS01.03.04
labeling the lengths
fourths of an inch. Show the data by
CCR.MD.B
making a line plot, where the horizontal
scale is marked off in appropriate units whole numbers, halves, or quarters.
Solve problems involving measurement and estimation of intervals of time, liquid
volumes, and masses of objects.
2.MD.11
Tell and write time to the nearest minute
CC.3.MD.1
Checking bus
and measure time intervals in minutes.
ESS01.03.01
schedules in a.m. and
Solve word problems involving addition
ESS01.03.05
p.m.
and subtraction of time intervals in
CCR.MD.B
minutes (e.g., by representing the
problem on a number line diagram).
2.MD.12
Measure and estimate liquid volumes and CC.3.MD.2
Following a recipe
masses of objects using standard units of ESS01.03.02
grams (g), kilograms (kg), and liters
CCR.MD.B
(l). Add, subtract, multiply, or divide to
solve one-step word problems involving
masses or volumes that are given in the
same units (e.g., by using drawings such
as a beaker with a measurement scale to
represent the problem).
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Geometric measurement: understand concepts of area and relate area to multiplication
and to addition.
2.MD.13 Recognize area as an attribute of plane
CC.3.MD.5
Buying carpeting, tiles
figures and understand concepts of area
ESS 01.03.04 or wallpaper
measurement.
CCR.MD.B
a. A square with side length 1 unit,
called “a unit square,” is said to
have “one square unit” of area, and
can be used to measure area.
b. A plane figure which can be
covered without gaps or overlaps
by n unit squares is said to have an
area of n square units.
2.MD.14 Measure areas by counting unit squares
CC.3.MD.6
(square cm, square m, square in, square
ESS 01.03.04
ft, and improvised units).
CCR.MD.B
2.MD.15 Relate area to the operations of
CC.3.MD.7
multiplication and addition.
ESS 01.03.02
a. Find the area of a rectangle with
ESS 01.03.04
whole-number side lengths by tiling CCR.MD.B
it, and show that the area is the
same as would be found by
multiplying the side lengths.
b. Multiply side lengths to find areas of
rectangles with whole-number side
lengths in the context of solving real
world and mathematical problems,
and represent whole-number
products as rectangular areas in
mathematical reasoning.
c. Use tiling to show in a concrete
case that the area of a rectangle
with whole-number side
lengths a and b + c is the sum
of a × b and a × c. Use area models
to represent the distributive
property in mathematical reasoning.
d. Recognize area as additive. Find
areas of rectilinear figures by
decomposing them into nonoverlapping rectangles and adding
the areas of the non-overlapping
parts, applying this technique to
solve real world problems.
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Geometric measurement: recognize perimeter as an attribute of plane figures and
distinguish between linear and area measures.
2.MD.16 Solve real world and mathematical
CC.3.MD.8
Figuring the amount
problems involving perimeters of polygons, ESS01.03.04
of molding needed
including finding the perimeter given the
ESS01.03.05
around a window
side lengths, finding an unknown side
CCR.MD.B
length, and exhibiting rectangles with the
same perimeter and different areas or with
the same area and different perimeters.
GEOMETRY (G)
Reason with shapes and their attributes.
2.G.1
Recognize and draw shapes having
specified attributes, such as a given
number of angles or a given number of
equal faces. Identify triangles,
quadrilaterals, pentagons, hexagons, and
cubes. Sizes are compared directly or
visually, not compared by measuring.
2.G.2
Partition a rectangle into rows and
columns of same-size squares and count
to find the total number of them.
2.G.3
Partition circles and rectangles into two,
three, or four equal shares, describe the
shares using the words halves, thirds, half
of, a third of, etc., and describe the whole
as two halves, three thirds, four fourths.
Recognize that equal shares of identical
wholes need not have the same shape.
2.G.4
Understand that shapes in different
categories (e.g., rhombuses, rectangles,
and others) may share attributes (e.g.,
having four sides), and that the shared
attributes can define a larger category
(e.g., quadrilaterals). Recognize
rhombuses, rectangles, and squares as
examples of quadrilaterals, and draw
examples of quadrilaterals that do not
belong to any of these subcategories.
2.G.5
Partition shapes into parts with equal
areas. Express the area of each part as a
unit fraction of the whole. For example,
partition a shape into 4 parts with equal
area, and describe the area of each part
as 1/4 of the area of the shape.
CC.2.G.1
ESS01.03.03
CCR.G.B
Creating a pattern for
laying tiles
CC.2.G.2
ESS01.03.02
Making flash cards
CC.2.G3
CCR.G.B
Cutting a pie or cake
into equal parts
CC.3.G.1
CCR.G.B
Drawing and
sculpting objects with
clay
CC.3.G.2
ESS01.03.02
CCR.G.B
Splitting a pizza or pie
into equal slices
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NRS Level 3 Overview
Operations and Algebraic Thinking (OA)
 Use the four operations with whole numbers to solve problems
 Gain familiarity with factors and multiples
 Generate and analyze patterns
 Write and interpret numerical expressions
 Analyze patterns and relationships
Number and Operations in Base Ten (NBT)
 Generalize place value understanding for multi-digit whole numbers
 Use place value understanding and properties of operations to perform multi-digit
arithmetic
 Understand the place value system
 Perform operations with multi-digit whole numbers and with decimals to
hundredths
Number and Operations – Fractions (NF)
 Extend understanding of fraction equivalence and ordering
 Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers
 Understand decimal notation for fractions, and compare decimal fractions
 Use equivalent fractions as a strategy to add and subtract fractions
 Apply and extend previous understandings of multiplication and division to
multiply and divide fractions
Measurement & Data (MD)
 Solve problems involving measurement and conversion of measurements from a
larger unit to a smaller unit
 Represent and interpret data
 Geometric measurement: understand concepts of angle and measure angles
 Convert like measurement units within a given measurement system
 Geometric measurement: understand concepts of volume and relate volume to
multiplication and to addition
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Illinois Community College Board, 5/2014
NRS Level 3 Overview (continued)
Geometry (G)
 Draw and identify lines and angles, and classify shapes by properties of their
lines and angles
 Graph points on the coordinate plane to solve real-world and mathematical
problems
 Classify two-dimensional figures into categories based on their properties
Mathematical Practices
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
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OVERVIEW EXPLANATION OF MATHEMATICS
NRS Level 3 – Low Intermediate Basic Education
(Grade Levels 4.0 – 5.9)
At the Low Intermediate Basic Education Level, instructional time should focus on six
critical areas18:
1. Developing understanding and fluency with multi-digit multiplication, and of
dividing to find quotients involving multi-digit dividends;
2. Developing an understanding of fraction equivalence, addition and subtraction of
fractions with like denominators and multiplication of fractions by whole numbers;
3. Developing fluency with addition and subtraction of fractions, and developing
understanding of the multiplication of fractions and of division of fractions in
limited cases (unit fractions divided by whole numbers and whole numbers
divided by unit fractions);
4. Extending division to 2-digit divisors, integrating decimal fractions into the place
value system and developing understanding of operations with decimals to
hundredths, and developing fluency with whole number and decimal operations;
5. Understanding that geometric figures can be analyzed and classified based on
their properties, such as having parallel sides, perpendicular sides, particular
angle measures, and symmetry; and
6. Developing understanding of volume.
(1) Adult learners generalize their understanding of place value to 1,000,000,
understanding the relative sizes of numbers in each place. They apply their
understanding of models for multiplication (equal-sized groups, arrays, area models),
place value, and properties of operations, in particular the distributive property, as they
develop, discuss, and use efficient, accurate, and generalizable methods to compute
products of multi-digit whole numbers. Depending on the numbers and the context, they
select and accurately apply appropriate methods to estimate or mentally calculate
products. They develop fluency with efficient procedures for multiplying whole numbers;
understand and explain why the procedures work based on place value and properties
of operations; and use them to solve problems. Students apply their understanding of
models for division, place value, properties of operations, and the relationship of division
to multiplication as they develop, discuss, and use efficient, accurate, and generalizable
procedures to find quotients involving multi-digit dividends. They select and accurately
apply appropriate methods to estimate and mentally calculate quotients, and interpret
remainders based upon the context.
(2) Adult learners develop understanding of fraction equivalence and operations with
fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and
they develop methods for generating and recognizing equivalent fractions. Students
extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and
18
Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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Illinois Community College Board, 5/2014
using the meaning of fractions and the meaning of multiplication to multiply a fraction by
a whole number.
(3) Adult learners apply their understanding of fractions and fraction models to represent
the addition and subtraction of fractions with unlike denominators as equivalent
calculations with like denominators. They develop fluency in calculating sums and
differences of fractions, and make reasonable estimates of them. Students also use the
meaning of fractions, of multiplication and division, and the relationship between
multiplication and division to understand and explain why the procedures for multiplying
and dividing fractions make sense. (Note: this is limited to the case of dividing unit
fractions by whole numbers and whole numbers by unit fractions.)
(4) Adult learners develop understanding of why division procedures work based on the
meaning of base-ten numerals and properties of operations. They finalize fluency with
multi-digit addition, subtraction, multiplication, and division. They apply their
understandings of models for decimals, decimal notation, and properties of operations
to add and subtract decimals to hundredths. They develop fluency in these
computations, and make reasonable estimates of their results. Students use the
relationship between decimals and fractions, as well as the relationship between finite
decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of
10 is a whole number), to understand and explain why the procedures for multiplying
and dividing finite decimals make sense. They compute products and quotients of
decimals to hundredths efficiently and accurately.
(5) Adult learners describe, analyze, compare, and classify two-dimensional shapes.
Through building, drawing, and analyzing two-dimensional shapes, students deepen
their understanding of properties of two-dimensional objects and the use of them to
solve problems involving symmetry.
(6) Adult learners recognize volume as an attribute of three-dimensional space. They
understand that volume can be measured by finding the total number of same-size units
of volume required to fill the space without gaps or overlaps. They understand that a 1unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select
appropriate units, strategies, and tools for solving problems that involve estimating and
measuring volume. They decompose three-dimensional shapes and find volumes of
right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.
They measure necessary attributes of shapes in order to determine volumes to solve
real world and mathematical problems.
The Standards for Mathematical Practice complement the content standards so that
students increasingly engage with the subject matter as they grow in mathematical
understanding and expertise.
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NRS Level 3 – Low Intermediate Basic Education
(Grade Levels 4.0 – 5.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This Skill
OPERATIONS AND ALGEBRAIC THINKING (OA)
Use the four operations with whole numbers to solve problems.
3.OA.1
Interpret a multiplication equation as a
CC.4.OA.1
comparison (e.g., interpret 35 = 5 x 7 as a
ESS01.03.02
statement that 35 is 5 times as many as 7
CCR.OA.C
and 7 times as many as 5). Represent verbal
statements of multiplicative comparisons as
multiplication equations.
3.OA.2
Multiply or divide to solve word problems
CC.4.OA.2
involving multiplicative comparison (e.g., by
ESS01.03.02
using drawings and equations with a symbol ESS01.03.05
for the unknown number to represent the
CCR.OA.C
problem), distinguishing multiplicative
comparison from additive comparison.
3.OA.3
Solve multistep word problems posed with
CC.4.OA.3
whole numbers and having whole-number
ESS01.03.02
answers using the four operations, including ESS01.03.05
problems in which remainders must be
CCR.OA.C
interpreted. Represent these problems using
equations with a letter standing for the
unknown quantity. Assess the
reasonableness of answers using mental
computation and estimation strategies
including rounding.
Gain familiarity with factors and multiples.
3.OA.4
Find all factor pairs for a whole number in the CC.4.OA.4
range 1-100. Recognize that a whole number ESS01.03.02
is a multiple of each of its factors. Determine CCR.OA.C
whether a given whole number in the range
1-100 is a multiple of a given one-digit
number. Determine whether a given whole
number in the range 1-100 is prime or
composite.
Calculating the total
number of items in
batches (e.g., 5
crates with 16
boxes to a crate)
Planning a
neighborhood party
Planning what kind
of pizza or
sandwiches to order
for an employee
luncheon
Figuring how many
ways change from
$100 can be
given/made
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Generate and analyze patterns.
3.OA.5
Generate a number or shape pattern that
follows a given rule. Identify apparent
features of the pattern that were not explicit
in the rule itself. For example, given the rule
“add 3” and the starting number 1, generate
terms in the resulting sequence and observe
that the terms appear to alternate between
odd and even numbers. Explain informally
why the numbers will continue to alternate in
this way.
Write and interpret numerical expressions.
3.OA.6
Use parentheses, brackets, or braces in
numerical expressions, and evaluate
expressions with these symbols.
3.OA.7
Write simple expressions that record
calculations with numbers, and interpret
numerical expressions without evaluating
them. For example, express the calculation
“add 8 and 7, then multiply by 2” as 2 × (8 +
7). Recognize that 3 × (18932 + 921) is three
times as large as 18932 + 921, without
having to calculate the indicated sum or
product.
Analyze patterns and relationships.
3.OA.8
Generate two numerical patterns using two
given rules. Identify apparent relationships
between corresponding terms. Form ordered
pairs consisting of corresponding terms from
the two patterns, and graph the ordered pairs
on a coordinate plane. For example, given
the rule “Add 3” and the starting number 0,
and given the rule “Add 6” and the starting
number 0, generate terms in the resulting
sequences, and observe that the terms in
one sequence are twice the corresponding
terms in the other sequence. Explain
informally why this is so.
CC.4.OA.5
ESS01.03.03
CCR.OA.C
Designing a
necklace and
describing the
assembly rule
CC.5.OA.1
ESS01.03.04
CCR.OA.C
CC.5.OA.2
ESS01.03.03
CCR.OA.C
Entering an
expression in a
spreadsheet
Creating and
solving word
problems
CC.5.OA.3
ESS01.03.06
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NUMBER AND OPERATIONS IN BASE TEN (NBT)
Generalize place value understanding for multi-digit whole numbers.
3.NBT.1
Recognize that in a multi-digit whole number, CC.4.NBT.1
Balancing a
a digit in one place represents ten times what ESS01.03.01
checkbook
it represents in the place to its right. For
ESS01.03.02
example, recognize that 700 ÷ 70 = 10 by
CCR.NBT.C
Determining a total
applying concepts of place value and
bill or expense
division.
3.NBT.2
Read and write multi-digit whole numbers
CC.4.NBT.2
Rounding numbers
using base-ten numerals, number names,
ESS01.03.03
to make
and expanded form. Compare two multi-digit CCR.NBT.C
approximate
numbers based on meanings of the digits in
calculations
each place, using >, =, and < symbols to
record the results of comparisons.
3.NBT.3
Use place value understanding to round
CC.4.NBT.3
multi-digit whole numbers to any place.
CCR.NBT.C
Use place value understanding and properties of operations to perform multi-digit
arithmetic.
3.NBT.4
Fluently add and subtract multi-digit whole
CC.4.NBT.4
Calculating down
numbers using the standard algorithm.
ESS01.03.02
and distance in
CCR.NBT.C
football
3.NBT.5
Multiply a whole number of up to four digits
CC.4.NBT.5
Checking delivery of
by a one-digit whole number, and multiply
ESS01.03.02
goods in small
two two-digit numbers, using strategies
CCR.NBT.C
batches
based on place value and the properties of
operations. Illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models.
3.NBT.6
Find whole-number quotients and
CC.4.NBT.6
Finding price of 2
remainders with up to four-digit dividends
ESS01.03.02
cartons of milk or 6
and one-digit divisors, using strategies based CCR.NBT.C
bottles of soda
on place value, the properties of operations,
and/or the relationship between multiplication
and division. Illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models.
Understand the place value system.
3.NBT.7
Recognize that in a multi-digit number, a digit CC.5.NBT.1
in one place represents 10 times as much as ESS01.03.01
it represents in the place to its right and 1/10 CCR.NBT.C
of what it represents in the place to its left.
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3.NBT.8
3.NBT.9
3.NBT.10
3.NBT.11
3.NBT.12
Explain patterns in the number of zeros of
the product when multiplying a number by
powers of 10, and explain patterns in the
placement of the decimal point when a
decimal is multiplied or divided by a power of
10. Use whole number exponents to denote
powers of 10.
Read, write, and compare decimals to
thousandths.
Read and write decimals to thousandths
using base-ten numerals, number names,
and expanded form (e.g., 347.392 = 3 × 100
+ 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) +
2 × (1/1000).
Compare two decimals to thousandths based
on meanings of the digits in each place,
using >, =, and < symbols to record the
results of comparisons.
Use place value understanding to round
decimals to any place.
CC.5.NBT.2
ESS01.03.02
CCR.NBT.C
CC.5.NBT.3
ESS01.03.01
CCR.NBT.C
CC.5.NBT.3a
ESS01.03.01
ESS01.03.02
CCR.NBT.C
CC.5.NBT.3b
ESS01.03.03
CCR.NBT.C
Using Dewey
Decimal System in
a library
CC.5.NBT.4
Understanding price
ESS01.03.01
tags or prices on a
CCR.NBT.C
menu
Perform operations with multi-digit whole numbers and with decimals to hundredths.
3.NBT.13 Fluently multiply multi-digit whole numbers
CC.5.NBT.5
Calculating
using the standard algorithm.
ESS01.03.02
attendance over
CCR.NBT.C
multiple events at
one location
3.NBT.14 Find whole-number quotients of whole
CC.5.NBT.6
Calculating miles
numbers with up to four-digit dividends and
ESS01.03.02
per gallon attained
two-digit divisors, using strategies based on
CCR.NBT.C
by a vehicle
place value, the properties of operations,
and/or the relationship between multiplication
and division. Illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models.
3.NBT.15 Add, subtract, multiply, and divide decimals
CC.5.NBT.7
Counting and
to hundredths, using concrete models or
ESS01.03.02
recording total value
drawings and strategies based on place
CCR.NBT.C
of an item
value, properties of operations, and/or the
purchased on sale
relationship between addition and
subtraction; relate the strategy to a written
method and explain the reasoning used.
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NUMBER & OPERATIONS – FRACTIONS (NF)
Extend understanding of fraction equivalence and ordering.
Explain why a fraction a/b is equivalent to a
3.NF.1
CC.4.NF.1
In recipe
fraction (n × a)/(n × b) by using visual fraction
ESS01.03.04 conversions,
models, with attention to how the number and
ESS01.03.05 determining equal
size of the parts differ even though the two
CCR.NF.C
amounts
fractions themselves are the same size. Use this
principle to recognize and generate equivalent
fractions.
3.NF.2
Compare two fractions with different numerators
CC.4.NF.2
In recipe
and different denominators (e.g., by creating
ESS01.03.03 conversions,
common denominators or numerators, or by
CCR.NF.C
determining equal
comparing to a benchmark fraction such as ½).
amounts (e.g.,
Recognize that comparisons are valid only when
tablespoons to
the two fractions refer to the same whole. Record
cups and cups to
the results of comparisons with symbols >, =, or
quarts)
<, and justify the conclusions (e.g., by using a
visual fraction model).
Build fractions from unit fractions by applying and extending previous understandings of
operations on whole numbers.
Understand a fraction a/b with a > 1 as a sum of
3.NF.3
CC.4.NF.3
Compare interest
fractions 1/b.
ESS01.03.02 rates (e.g., 1 ¼%
a. Understand addition and subtraction of
ESS01.03.03 versus 1 ½%)
fractions as joining and separating parts
ESS01.03.04
referring to the same whole.
CCR.NF.C
b. Decompose a fraction into a sum of
fractions with the same denominator in
more than one way, recording each
decomposition by an equation. Justify
decompositions (e.g., by using a visual
fraction model). Examples: 3/8 = 1/8 + 1/8
+ 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 =
8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like
denominators (e.g., by replacing each
mixed number with an equivalent fraction,
and/or by using properties of operations
and the relationship between addition and
subtraction).
d. Solve word problems involving addition
and subtraction of fractions referring to the
same whole and having like denominators
(e.g., by using visual fraction models and
equations to represent the problem).
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3.NF.4
Apply and extend previous understandings of
CC.4.NF.4
Planning a
multiplication to multiply a fraction by a whole
ESS01.03.02 neighborhood
number.
ESS01.03.04 party
a. Understand a fraction a/b as a
CCR.NF.C
multiple of 1/b. For example, use a visual
fraction model to represent 5/4 as the
Generating
product 5 × (1/4), recording the conclusion
solutions using
by the equation 5/4 = 5 × (1/4).
mental
b. Understand a multiple of a/b as a multiple
mathematics in
of 1/b, and use this understanding to
situations
multiply a fraction by a whole number. For
involving common
example, use a visual fraction model to
unit fractions
express 3 × (2/5) as 6 × (1/5), recognizing
this product as 6/5. (In general, n × (a/b) =
(n × a)/b.)
c. Solve word problems involving
multiplication of a fraction by a whole
number (e.g., by using visual fraction
models and equations to represent the
problem). For example, if each person at a
party will eat 3/8 of a pound of roast beef,
and there will be 5 people at the party,
how many pounds of roast beef will be
needed? Between what two whole
numbers does your answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
3.NF.5
Express a fraction with denominator 10 as an
CC.4.NF.5
equivalent fraction with denominator 100, and
ESS01.03.01
use this technique to add two fractions with
respective denominators 10 and 100. For
example, express 3/10 as 30/100 and add 3/10 +
4/100 = 34/100.
3.NF.6
Use decimal notation for fractions with
CC.4.NF.6
Understanding
denominators 10 or 100. For example, rewrite
ESS01.03.01 how the scale
0.62 as 62/100; describe a length as 0.62 meters; CCR.NF.C
works at the deli
locate 0.62 on a number line diagram.
counter
3.NF.7
Compare two decimals to the hundredths place
CC.4.NF.7
Reading and
by reasoning about their size. Recognize that
ESS01.03.03 comparing gas
comparisons are valid only when two decimals
CCR.NF.C
prices
refer to the same whole. Record the results of
comparisons with the symbols >, =, or <, and
Reading and
justify the conclusions (e.g., by using a visual
comparing metric
model).
measurements
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Use equivalent fractions as a strategy to add and subtract fractions.
3.NF.8
Add and subtract fractions with unlike
CC.5.NF.1
Understanding
denominators (including mixed numbers) by
ESS01.03.01 how much food is
replacing given fractions with equivalent fractions ESS01.03.02 left after a party
in such a way as to produce an equivalent sum or CCR.NF.C
difference of fractions with like denominators. For
example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In
general, a/b + c/d = (ad + bc)/bd.)
3.NF.9
Solve word problems involving addition and
CC.5.NF.2
Understanding
subtraction of fractions referring to the same
ESS01.03.04 how to read a
whole, including cases of unlike denominators
CCR.NF.C
digital scale when
(e.g., by using visual fraction models or equations
placing a fraction
to represent the problem). Use benchmark
order at the deli
fractions and number sense of fractions to
estimate mentally and assess the
reasonableness of answers. For example,
recognize an incorrect result 2/5 + 1/2 = 3/7 by
observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
3.NF.10 Interpret a fraction as division of the numerator
CC.5.NF.3
Planning what
by the denominator (a/b = a ÷ b). Solve word
ESS01.03.03 kind of pizza or
problems involving division of whole numbers
ESS01.03.04 sandwiches to
leading to answers in the form of fractions or
CCR.NF.C
order for an
mixed numbers (e.g., by using visual fraction
employee
models or equations to represent the problem).
luncheon
For example, interpret 3/4 as the result of dividing
3 by 4, noting that 3/4 multiplied by 4 equals 3
and that when 3 wholes are shared equally
among 4 people, each person has a share of size
3/4. If 9 people want to share a 50-pound sack of
rice equally by weight, how many pounds of rice
should each person get? Between what two
whole numbers does your answer lie?
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Illinois Community College Board, 5/2014
3.NF.11
3.NF.12
3.NF.13
Apply and extend previous understandings of
multiplication to multiply a fraction or whole
number by a fraction.
a. Interpret the product (a/b) × q as a parts
of a partition of q into b equal parts;
equivalently, as the result of a sequence of
operations a × q ÷ b. For example, use a
visual fraction model to show (2/3) x 4 =
8/3, and create a story context for this
equation. Do the same with (2/3) x (4/5) =
8/15. (In general, (a/b) x (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional
side lengths by tiling it with unit squares of
the appropriate unit fraction side lengths,
and show that the area is the same as
would be found by multiplying the side
lengths. Multiply fractional side lengths to
find areas of rectangles, and represent
fraction products as rectangular areas.
Interpret multiplication as scaling (resizing) by:
a. Comparing the size of a product to the
size of one factor on the basis of the size
of the other factor, without performing the
indicated multiplication.
b. Explaining why multiplying a given number
by a fraction greater than 1 results in a
product greater than the given number
(recognizing multiplication by whole
numbers greater than 1 as a familiar
case); explaining why multiplying a given
number by a fraction less than 1 results in
a product smaller than the given number;
and relating the principle of fraction
equivalence a/b = (n×a) / (n×b) to the
effect of multiplying a/b by 1.
Solve real world problems involving multiplication
of fractions and mixed numbers (e.g., by using
visual fraction models or equations to represent
the problem).
CC.5.NF.4
ESS01.03.04
CCR.NF.C
CC.5.NF.5
ESS01.03.04
ESS01.03.05
CCR.NF.C
CC.5.NF.6
ESS01.03.04
CCR.NF.C
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3.NF.14
Apply and extend previous understandings of
CC.5.NF.7
division to divide unit fractions by whole numbers ESS01.03.04
and whole numbers by unit fractions
CCR.NF.C
a. Interpret division of a unit
fraction by a non-zero whole number, and
compute such quotients. For example,
create a story context for (1/3) ÷ 4 and use
a visual fraction model to show the
quotient. Use the relationship between
multiplication and division to explain that
(1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a
unit fraction, and compute such quotients.
For example, create a story context for 4 ÷
(1/5) and use a visual fraction model to
show the quotient. Use the relationship
between multiplication and division to
explain that 4 ÷ (1/5) = 20 because 20 ×
(1/5) = 4.
c. Solve real-world problems involving
division of unit fractions by non-zero whole
numbers and division of whole numbers by
unit fractions (e.g., by using visual fraction
models and equations to represent the
problem). For example, how much
chocolate will each person get if 3 people
share a 1/2 pound of chocolate equally?
How many 1/3-cup servings are in 2 cups
of raisins?
MEASUREMENT & DATA (MD)
Solve problems involving measurement and conversion of measurements from a larger
unit to a smaller unit.
3.MD.1
Know relative sizes of measurement units
CC.4.MD.1
Following the
within one system of units including km, m, cm; ESS01.03.04 growth in height or
kg, g; lb., oz.; l, ml; hr., min, sec. Within a single
weight of babies,
system of measurement, express
children, or plants
measurements in a larger unit in terms of a
smaller unit. Record measurement equivalents
in a two-column table. For example: Know that
1 ft is 12 times as long as 1 in. Express the
length of a 4 ft snake as 48 in. Generate a
conversion table for feet and inches listing the
number pairs (1, 12), (2, 24), (3, 36).
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3.MD.2
Use the four operations to solve word problems
involving distances, intervals of time, liquid
volumes, masses of objects, and money,
including problems involving simple fractions or
decimals, and problems that require expressing
measurements given in a larger unit in terms of
a smaller unit. Represent measurement
quantities using diagrams such as number line
diagrams that feature a measurement scale.
3.MD.3
Apply the area and perimeter formulas for
rectangles in real world and mathematical
problems. For example, find the width of a
rectangular room given the area of the flooring
and the length, by viewing the area formula as
a multiplication equation with an unknown
factor.
Represent and interpret data.
3.MD.4
Make a line plot to display a data set of
measurements in fractions of a unit (1/2, 1/4,
1/8). Solve problems involving addition and
subtraction of fractions by using information
presented in line plots. For example, from a line
plot find and interpret the difference in length
between the longest and shortest specimens in
an insect collection.
3.MD.5
Make a line plot to display a data set of
measurements in fractions of a unit (1/2, 1/4,
1/8). Use operations on fractions for this grade
to solve problems involving information
presented in line plots. For example, given
different measurements of liquid in identical
beakers, find the amount of liquid each beaker
would contain if the total amount in all the
beakers were redistributed equally.
CC.4.MD.2
ESS01.03.06
CCR.MD.C
Planning a trip
using distance
and time from an
atlas legend
CC.4.MD.3
CCR.MD.C
Finding the length
of fencing around
a garden
Buying weatherstripping
CC.4.MD.4
ESS01.03.06
CCR.MD.C
Calculating the
size of a container
required to hold
items of various
lengths
CC.5.MD.2
ESS01.03.06
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Geometric measurement: understand concepts of angle and measure angles.
3.MD.6
Recognize angles as geometric shapes that
CC.4.MD.5
are formed wherever two rays share a
ESS01.03.04
common endpoint, and understand concepts CCR.MD.C
of angle measurement:
a. An angle is measured with reference
to a circle with its center at the
common endpoint of the rays, by
considering the fraction of the circular
arc between the points where the two
rays intersect the circle. An angle that
turns through 1/360 of a circle is
called a “one-degree angle,” and can
be used to measure angles.
b. An angle that turns through in onedegree angles is said to have an
angle measure of n degrees.
3.MD.7
Measure angles in whole-number degrees
CC.4.MD.6
Checking the angle
using a protractor. Sketch angles of specified ESS01.03.04
of incline of a road
measure.
CCR.MD.C
or wheelchair ramp
3.MD.8
Recognize angle measure as additive. When CC.4.MD.7
an angle is decomposed into nonESS01.03.02
overlapping parts, the angle measure of the
ESS01.03.04
whole is the sum of the angle measures of
CCR.MD.C
the parts. Solve addition and subtraction
problems to find unknown angles on a
diagram in real world and mathematical
problems (e.g., by using an equation with a
symbol for the unknown angle measure).
Convert like measurement units within a given measurement system.
3.MD.9
Convert among different-sized standard
CC.5.MD.1
Doing home repairs
measurement units within a given
ESS01.03.04
and carpentry
measurement system (e.g., convert 5 cm to
CCR.MD.C
projects
0.05 m), and use these conversions in
solving multi-step real world problems.
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Geometric measurement: understand concepts of volume and relate volume to
multiplication and to addition.
3.MD.10
Recognize volume as an attribute of solid
CC.5.MD.3
figures and understand concepts of volume
CCR.MD.C
measurement.
a. A cube with side length 1 unit, called
a “unit cube,” is said to have “one
cubic unit” of volume, and can be
used to measure volume.
b. A solid figure which can be packed
without gaps or overlaps using n unit
cubes is said to have a volume of n
cubic units.
3.MD.11
Measure volumes by counting unit cubes,
CC.5.MD.4
using cubic cm, cubic in, cubic ft, and
ESS01.03.04
improvised units.
CCR.MD.C
3.MD.12
Relate volume to the operations of
CC.5.MD.5
Determining the
multiplication and addition and solve real
ESS01.03.02
amount of sand
world and mathematical problems involving
ESS01.03.04
needed to fill a
volume.
CCR.MD.C
sandbox
a. Find the volume of a right rectangular
prism with whole-number side lengths
Determining the
by packing it with unit cubes, and
amount of mulch
show that the volume is the same as
needed to complete
would be found by multiplying the
a landscaping
edge lengths, equivalently by
project
multiplying the height by the area of
the base. Represent three-fold wholeDetermining the
number products as volumes (e.g., to
amount of water
represent the associative property of
needed to fill a pool
multiplication.
b. Apply the formulas V =(l)(w)(h) and V
= (b)(h) for rectangular prisms to find
volumes of right rectangular prisms
with whole-number edge lengths in
the context of solving real world and
mathematical problems.
c. Recognize volume as additive. Find
volumes of solid figures composed of
two non-overlapping right rectangular
prisms by adding the volumes of the
non-overlapping parts, applying this
technique to solve real world
problems.
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GEOMETRY (G)
Draw and identify lines and angles, and classify shapes by properties of their lines and
angles.
3.G.1
Draw points, lines, line segments, rays,
CC.4.G.1
Creating designs
angles (right, acute, obtuse), and
CCR.G.C
perpendicular and parallel lines. Identify
these in two-dimensional figures.
3.G.2
Classify two-dimensional figures based on
CC.4.G.2
Writing certain
the presence or absence of parallel or
letters (e.g., A, C, D,
perpendicular lines, or the presence or
E, F,H, L,O,T, etc.)
absence of angles of a specified size.
Recognize right triangles as a category, and
identify right triangles.
3.G.3
Recognize a line of symmetry for a twoCC.4.G.3
Creating holiday
dimensional figure as a line across the figure
designs for greeting
such that the figure can be folded along the
cards or crafts
line into matching parts. Identify linesymmetric figures and draw lines of
symmetry.
Graph points on the coordinate plane to solve real-world and mathematical problems.
3.G.4
Use a pair of perpendicular number lines,
CC.5.G.1
called axes, to define a coordinate system,
ESS01.03.04
with the intersection of the lines (the origin)
CCR.G.C
arranged to coincide with the 0 on each line
and a given point in the plane located by
using an ordered pair of numbers, called its
coordinates. Understand that the first number
indicates how far to travel from the origin in
the direction of one axis, and the second
number indicates how far to travel in the
direction of the second axis, with the
convention that the names of the two axes
and the coordinates correspond (e.g., x-axis
and x-coordinate, y-axis and y-coordinate).
3.G.5
Represent real world and mathematical
CC.5.G.2
problems by graphing points in the first
ESS01.03.04
quadrant of the coordinate plane, and
CCR.G.C
interpret coordinate values of points in the
context of the situation.
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Classify two-dimensional figures into categories based on their properties.
3.G.6
Understand that attributes belonging to a
CC.5.G.3
Sorting shapes of all
category of two-dimensional figures also
CCR.G.C
sizes into subbelong to all subcategories of that category.
categories and
For example, all rectangles have four right
explaining why
angles and squares are rectangles, so all
squares have four right angles.
3.G.7
Classify two-dimensional figures in a
CC.5.G.4
hierarchy based on properties.
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NRS Level 4 Overview
Ratios and Proportional Relationships (RP)
 Understand ratio concepts and use ratio reasoning to solve problems
 Analyze proportional relationships and use them to solve real-world and
mathematical problems
The Number System (NS)
 Apply and extend previous understandings of multiplication and division to divide
fractions by fractions
 Compute fluently with multi-digit numbers and find common factors and multiples
 Apply and extend previous understandings of numbers to the system of rational
numbers
 Apply and extend previous understandings of operations with fractions to add,
subtract, multiply, and divide rational numbers
 Know that there are numbers that are not rational, and approximate them by
rational numbers
Expressions and Equations (EE)
 Apply and extend previous understandings of arithmetic to algebraic expressions
 Reason about and solve one-variable equations and inequalities
 Represent and analyze quantitative relationships between dependent and
independent variables
 Use properties of operations to generate equivalent expressions
 Solve real-life and mathematical problems using numerical and algebraic
expressions and equations
 Work with radicals and integer exponents
 Understand the connections between proportional relationships, lines, and linear
equations
 Analyze and solve linear equations and pairs of simultaneous linear equations
Functions (F)
 Define, evaluate, and compare functions
 Use functions to model relationships between quantities
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NRS Level 4 Overview (continued)
Geometry (G)
 Solve real-world and mathematical problems
involving area, surface area, and volume
 Draw, construct, and describe geometrical
figures and describe the relationship between
them
 Solve real-life and mathematical problems
involving angle measure, area, surface area,
and volume
 Understand congruence and similarity using
physical models, transparencies, or geometry
software
 Understand and apply the Pythagorean
Theorem
 Solve real-world and mathematical problems
involving volume of cylinders, cones, and
spheres
Statistics and Probability (SP)
 Develop understanding of statistical variability
 Summarize and describe distributions
 Use random sampling to draw inferences about
a population
 Draw informal comparative inferences about
two populations
 Investigate chance processes and develop,
use, and evaluate probability models
 Investigate patterns of association in bivariate
data
Mathematical Practices
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
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OVERVIEW EXPLANATION OF MATHEMATICS
NRS Level 4 – High Intermediate Basic Education
(Grade Levels 6.0 – 8.9)
At the High Intermediate Basic Education Level, instructional time should focus on
seven critical area19s:
1. Developing understanding of and applying proportional relationships.
2. Developing understanding of operations with rational numbers and working with
expressions and linear equations.
3. Formulating and reasoning about expressions and equations, including modeling
an association in bivariate data with a linear equation, and solving linear
equations and systems of linear equations.
4. Grasping the concept of a function and using functions to describe quantitative
relationships.
5. Solving problems involving scale drawings and informal geometric constructions,
and working with two- and three-dimensional shapes to solve problems involving
area, surface area, and volume.
6. Analyzing two- and three-dimensional space and figures using distance, angle,
similarity, and congruence, and understanding and applying the Pythagorean
Theorem.
7. Drawing inferences about populations based on samples, they build on their
previous work with single data distributions to compare two data distributions and
address questions about differences between populations.
(1) Adult learners extend their understanding of ratios and develop understanding of
proportionality to solve single- and multi-step problems. They use their understanding of
ratios and proportionality to solve a wide variety of percent problems, including those
involving discounts, interest, taxes, tips, and percent increase or decrease. Learners
solve problems about scale drawings by relating corresponding lengths between the
objects or by using the fact that relationships of lengths within an object are preserved
in similar objects; they also graph proportional relationships and understand the unit
rate informally as a measure of the steepness of the related line, called the slope. They
distinguish proportional relationships from other relationships.
(2) Adult learners develop a unified understanding of number, recognizing fractions,
decimals (that have a finite or a repeating decimal representation), and percents as
different representations of rational numbers. Learners extend addition, subtraction,
multiplication, and division to all rational numbers, maintaining the properties of
operations and the relationships between addition and subtraction, and multiplication
and division. By applying these properties, and by viewing negative numbers in terms of
everyday contexts (e.g., amounts owed or temperatures below zero), they explain and
interpret the rules for adding, subtracting, multiplying, and dividing with negative
numbers. They use the arithmetic of rational numbers as they formulate expressions
and equations in one variable and use these equations to solve problems.
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Common Core State Standards for Mathematics, 2010, www.corestandards.org.
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(3) Adult learners use linear equations and systems of linear equations to represent,
analyze, and solve a variety of problems. They recognize equations for proportions (y/x
= m or y = mx) as special linear equations (y = mx + b), understanding that the constant
of proportionality (m) is the slope, and the graphs are lines through the origin. They
understand that the slope (m) of a line is a constant rate of change, so that if the input
or x-coordinate changes by an amount A, the output or y-coordinate changes by the
amount m·A. Learners also use a linear equation to describe the association between
two quantities in bivariate data (such as arm span vs. height for students in a
classroom). At the conclusion of Level 4, fitting the model and assessing its fit to the
data are done informally. Interpreting the model in the context of the data requires
students to express a relationship between the two quantities in question and to
interpret components of the relationship (such as slope and y-intercept) in terms of the
situation. Adult learners strategically choose and efficiently implement procedures to
solve linear equations in one variable, understanding that when they use the properties
of equality and the concept of logical equivalence, they maintain the solutions of the
original equation. Learners solve systems of two linear equations in two variables and
relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the
same line. They use linear equations, systems of linear equations, linear functions, and
their understanding of slope of a line to analyze situations and solve problems.
(4) Adult learners grasp the concept of a function as a rule that assigns to each input
exactly one output. They understand that functions describe situations where one
quantity determines another. They can translate among representations and partial
representations of functions (noting that tabular and graphical representations may be
partial representations), and they describe how aspects of the function are reflected in
the different representation.
(5) Adult learners continue their work on solving problems involving the area and
circumference of a circle and surface area of three-dimensional objects. In preparation
for work on congruence and similarity, they reason about relationships among twodimensional figures using scale drawings and informal geometric constructions, and
they gain familiarity with the relationships between angles formed by intersecting lines.
Learners work with three-dimensional figures, relating them to two-dimensional figures
by examining cross-sections. They solve real-world and mathematical problems
involving area, surface area, and volume of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes and right prisms.
(6) Adult learners use ideas about distance and angles, how they behave under
translations, rotations, reflections, and dilations, and ideas about congruence and
similarity to describe and analyze two-dimensional figures and to solve problems. They
show that the sum of the angles in a triangle is the angle formed by a straight line and
that various configurations of lines give rise to similar triangles because of the angles
created when a transversal cuts parallel lines. Learners understand the statement of the
Pythagorean Theorem and its converse, and can explain why the Pythagorean
Theorem holds, for example, by decomposing a square in two different ways. Finally,
learners apply the Pythagorean Theorem to find distances between points on the
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coordinate plane, to find lengths, and to analyze polygons. Students complete their work
on volume by solving problems involving cones, cylinders, and spheres.
(7) Adult learners begin informal work with random sampling to generate data sets and
learn about the importance of representative samples for drawing inferences.
The Standards for Mathematical Practice complement the content standards so that
students increasingly engage with the subject matter as they grow in mathematical
understanding and expertise.
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NRS Level 4 – High Intermediate Basic Education
(Grade Levels 6.0-8.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This Skill
RATIOS AND PROPORTIONAL RELATIONSHIPS (RP)
Understand ratio concepts and use ratio reasoning to solve problems.
4.RP.1
Understand the concept of a ratio and use ratio CC.6.RP.1
language to describe a ratio relationship
ESS01.03.04
between two quantities. For example, “The
CCR.RP.C
ratio of wings to beaks in the bird house at the
zoo was 2:1, because for every 2 wings there
was 1 beak.” “For every vote candidate A
received, candidate C received nearly three
votes.”
Understand the concept of a unit rate a/b
4.RP.2
CC.6.RP.2
associated with a ratio a:b with b ≠ 0 (b not
ESS01.03.04
equal to zero), and use rate language in the
CCR.RP.C
context of a ratio relationship. For example,
"This recipe has a ratio of 3 cups of flour to 4
cups of sugar, so there is 3/4 cup of flour for
each cup of sugar." "We paid $75 for 15
hamburgers, which is a rate of $5 per
hamburger." (Expectations for unit rates in this
grade are limited to non-complex fractions.)
Calculating miles
per gallon attained
by a vehicle
Estimating travel
time in hours based
on distance and
speed
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4.RP.3
Use ratio and rate reasoning to solve realCC.6.RP.3
Calculating miles
world and mathematical problems (e.g., by
ESS01.03.06
per gallon attained
reasoning about tables of equivalent ratios,
CCR.RP.D
by a vehicle
tape diagrams, double number line diagrams,
or equations).
a. Make tables of equivalent ratios relating
quantities with whole-number
measurements, find missing values in
the tables, and plot the pairs of values
on the coordinate plane. Use tables to
compare ratios.
b. Solve unit rate problems including those
involving unit pricing and constant
speed. For example, if it took 7 hours to
mow 4 lawns, then at that rate, how
many lawns could be mowed in 35
hours? At what rate were lawns being
mowed?
c. Find a percent of a quantity as a rate
per 100 (e.g., 30% of a quantity means
30/100 times the quantity); solve
problems involving finding the whole
given a part and the percent.
d. Use ratio reasoning to convert
measurement units; manipulate and
transform units appropriately when
multiplying or dividing quantities.
Analyze proportional relationships and use them to solve real-world and mathematical
problems.
4.RP.4
Compute unit rates associated with ratios of
CC.7.RP.1
Mixing various
fractions, including ratios of lengths, areas and ESS01.03.04
quantities of
other quantities measured in like or different
ESS01.03.05
cleaning fluids
units. For example, if a person walks 1/2 mile
CCR.RP.D
based on one set of
in each 1/4 hour, compute the unit rate as the
directions
complex fraction (1/2)/(1/4) miles per hour,
equivalent to 2 miles per hour.
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4.RP.5
4.RP.6
Recognize and represent proportional
relationships between quantities.
a. Decide whether two quantities are in a
proportional relationship (e.g., by testing
for equivalent ratios in a table or
graphing on a coordinate plane and
observing whether the graph is a
straight line through the origin).
b. Identify the constant of proportionality
(unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of
proportional relationships.
c. Represent proportional relationships by
equations. For example, if total cost t is
proportional to the number n of items
purchased at a constant price p, the
relationship between the total cost and
the number of items can be expressed
as t = pn.
d. Explain what a point (x, y) on the graph
of a proportional relationship means in
terms of the situation, with special
attention to the points (0, 0) and (1, r)
where r is the unit rate.
Use proportional relationships to solve
multistep ratio and percent problems.
Examples: simple interest, tax, markups and
markdowns, gratuities and commissions, fees,
percent increase and decrease, percent error.
CC.7.RP.2
ESS01.03.06
CCR.RP.D
Charting image size
and projector
distance from wall or
screen and making
predictions about
untested values
Calculating the
proper distance to
place a projector
from its screen to
achieve a particular
image size
CC.7.RP.3
ESS01.03.02
CCR.RP.D
Determining the
savings when
buying something
50% off
Determining the
difference in tip at
15% or 20% of bill
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THE NUMBER SYSTEM (NS)
Apply and extend previous understandings of multiplication and division to divide
fractions by fractions.
4.NS.1
Interpret and compute quotients of fractions,
CC.6.NS.1
Preparing a recipe
and solve word problems involving division of
ESS01.03.04
for a smaller group
fractions by fractions (e.g., by using visual
CCR.NS.C
than the intended
fraction models and equations to represent the
group
problem). For example, create a story context
for (2/3) ÷ (3/4) and use a visual fraction model
to show the quotient; use the relationship
between multiplication and division to explain
that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is
2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How
much chocolate will each person get if 3
people share 1/2 lb of chocolate equally? How
many 3/4-cup servings are in 2/3 of a cup of
yogurt? How wide is a rectangular strip of land
with length 3/4 mi and area 1/2 square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
4.NS.2
Fluently divide multi-digit numbers using the
CC.6.NS.2
standard algorithm.
CCR.NS.C
4.NS.3
Fluently add, subtract, multiply, and divide
CC.6.NS.3
Using a four function
multi-digit decimals using the standard
CCR.NS.C
calculator to find
algorithm for each operation.
hourly rate given
weekly pay or vice
versa
4.NS.4
Find the greatest common factor of two whole
CC.6.NS.4
numbers less than or equal to 100 and the
CCR.NS.C
least common multiple of two whole numbers
less than or equal to 12. Use the distributive
property to express a sum of two whole
numbers 1–100 with a common factor as a
multiple of a sum of two whole numbers with
no common factor. For example, express 36 +
8 as 4 (9 + 2).
Apply and extend previous understandings of numbers to the system of rational numbers.
4.NS.5
Understand that positive and negative
CC.6.NS.5
Understanding wind
numbers are used together to describe
CCR.NS.D
chill information
quantities having opposite directions or values
(e.g., temperature above/below zero, elevation
Staying “in the
above/below sea level, debits/credits,
black” or going “into
positive/negative electric charge); use positive
the red” when
and negative numbers to represent quantities
paying bills
in real-world contexts, explaining the meaning
of 0 in each situation.
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4.NS.6
Understand a rational number as a point on
CC.6.NS.6
the number line. Extend number line diagrams CCR.NS.D
and coordinate axes familiar from previous
grades to represent points on the line and in
the plane with negative number coordinates.
a. Recognize opposite signs of numbers
as indicating locations on opposite sides
of 0 on the number line; recognize that
the opposite of the opposite of a
number is the number itself (e.g., –(–3)
= 3, and that 0 is its own opposite).
b. Understand signs of numbers in ordered
pairs as indicating locations in
quadrants of the coordinate plane;
recognize that when two ordered pairs
differ only by signs, the locations of the
points are related by reflections across
one or both axes.
c. Find and position integers and other
rational numbers on a horizontal or
vertical number line diagram; find and
position pairs of integers and other
rational numbers on a coordinate plane.
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4.NS.7
4.NS.8
Understand ordering and absolute value of
rational numbers.
a. Interpret statements of inequality as
statements about the relative position of
two numbers on a number line diagram.
For example, interpret –3 > –7 as a
statement that –3 is located to the right
of –7 on a number line oriented from left
to right.
b. Write, interpret, and explain statements
of order for rational numbers in realworld contexts. For example, write –3°C
> –7°C to express the fact that –3°C is
warmer than –7°C.
c. Understand the absolute value of a
rational number as its distance from 0
on the number line; interpret absolute
value as magnitude for a positive or
negative quantity in a real-world
situation. For example, for an account
balance of –30 dollars, write |–30| = 30
to describe the size of the debt in
dollars.
d. Distinguish comparisons of absolute
value from statements about order. For
example, recognize that an account
balance less than –30 dollars
represents a debt greater than 30
dollars.
Solve real-world and mathematical problems
by graphing points in all four quadrants of the
coordinate plane. Include use of coordinates
and absolute value to find distances between
points with the same first coordinate or the
same second coordinate.
CC.6.NS.7
ESS01.03.03
ESS01.03.04
CCR.NS.D
Reading
thermometers
Reading a bank
balance
Balancing a check
book
CC.6.NS.8
CCR.NS.D
Tracking
temperature and
precipitation rates
throughout the year
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Apply and extend previous understandings of operations with fractions to add, subtract,
multiply, and divide rational numbers.
4.NS.9
Apply and extend previous understandings of
CC.7.NS.1
addition and subtraction to add and subtract
CCR.NS.D
rational numbers; represent addition and
subtraction on a horizontal or vertical number
line diagram.
a. Describe situations in which opposite
quantities combine to make 0. For
example, a hydrogen atom has 0
charge because its two constituents are
oppositely charged.
b. Understand p + q as the number
located a distance |q| from p, in the
positive or negative direction depending
on whether q is positive or negative.
Show that a number and its opposite
have a sum of 0 (are additive inverses).
Interpret sums of rational numbers by
describing real-world contexts.
c. Understand subtraction of rational
numbers as adding the additive inverse,
p – q = p + (–q). Show that the distance
between two rational numbers on the
number line is the absolute value of
their difference, and apply this principle
in real-world contexts.
d. Apply properties of operations as
strategies to add and subtract rational
numbers.
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4.NS.10
Apply and extend previous understandings of
CC.7.NS.2
multiplication and division and of fractions to
ESS01.03.02
multiply and divide rational numbers.
CCR.NS.D
a. Understand that multiplication is
extended from fractions to rational
numbers by requiring that operations
continue to satisfy the properties of
operations, particularly the distributive
property, leading to products such as (–
1)(–1) = 1 and the rules for multiplying
signed numbers. Interpret products of
rational numbers by describing realworld contexts.
b. Understand that integers can be
divided, provided that the divisor is not
zero, and every quotient of integers
(with non-zero divisor) is a rational
number. If p and q are integers then –
(p/q) = (–p)/q = p/(–q). Interpret
quotients of rational numbers by
describing real-world contexts.
c. Apply properties of operations as
strategies to multiply and divide rational
numbers.
d. Convert a rational number to a decimal
using long division; know that the
decimal form of a rational number
terminates in 0s or eventually repeats.
4.NS.11 Solve real-world and mathematical problems
CC.7.NS.3
involving the four operations with rational
ESS01.03.02
numbers. (Computations with rational numbers CCR.NS.D
extend the rules for manipulating fractions to
complex fractions.)
Know that there are numbers that are not rational, and approximate them by rational
numbers.
4.NS.12 Understand informally that every number has a CC.8.NS.1
decimal expansion; the rational numbers are
those with decimal expansions that terminate
in 0's or eventually repeat. Know that other
numbers are called irrational.
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4.NS.13
Use rational approximations of irrational
CC.8.NS.2
numbers to compare the size of irrational
ESS01.03.06
numbers, locate them approximately on a
CCR.NS.D
number line diagram, and estimate the value of
expressions (e.g., π2). For example, by
truncating the decimal expansion of √2, show
that √2 is between 1 and 2, then between 1.4
and 1.5, and explain how to continue on to get
better approximations.
EXPRESSIONS AND EQUATIONS (EE)
Apply and extend previous understandings of arithmetic to algebraic expressions.
4.EE.1
Write and evaluate numerical expressions
CC.6.EE.1
Entering an
involving whole-number exponents
CCR.EE.C
expression in a
spreadsheet
4.EE.2
Write, read, and evaluate expressions in which CC.6.EE.2
Keeping personal
letters stand for numbers.
ESS01.03.04
finance records
a. Write expressions that record
CCR.EE.C
operations with numbers and with
Converting
letters standing for numbers. For
temperature
example, express the calculation
between Celsius
“subtract y from 5” as 5 – y.
and Fahrenheit
b. Identify parts of an expression using
mathematical terms (sum, term,
Finding interest on a
product, factor, quotient, coefficient);
loan from a table
view one or more parts of an expression
as a single entity. For example,
describe the expression 2(8 + 7) as a
product of two factors; view (8 + 7) as
both a single entity and a sum of two
terms.
c. Evaluate expressions at specific values
for their variables. Include expressions
that arise from formulas in real-world
problems. Perform arithmetic
operations, including those involving
whole-number exponents, in the
conventional order when there are no
parentheses to specify a particular order
(order of operations). For example, use
the formulas V = s3 and A = 6 s2 to find
the volume and surface area of a cube
with sides of length s = 1/2.
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4.EE.3
Apply the properties of operations to generate CC.6.EE.3
equivalent expressions. For example, apply
ESS01.03.04
the distributive property to the expression 3(2
CCR.EE.C
+ x) to produce the equivalent expression 6 +
3x; apply the distributive property to the
expression 24x + 18y to produce the
equivalent expression 6 (4x + 3y); apply
properties of operations to y + y + y to produce
the equivalent expression 3y.
4.EE.4
Identify when two expressions are equivalent
CC.6.EE.4
(i.e., when the two expressions name the
CCR.EE.C
same number regardless of which value is
substituted into them). For example, the
expressions y + y + y and 3y are equivalent
because they name the same number
regardless of which number y stands for.
Reason about and solve one-variable equations and inequalities.
4.EE.5
Understand solving an equation or inequality
CC.6.EE.5
as a process of answering a question: which
CCR.EE.C
values from a specified set, if any, make the
equation or inequality true? Use substitution to
determine whether a given number in a
specified set makes an equation or inequality
true.
4.EE.6
Use variables to represent numbers and write
CC.6.EE.6
expressions when solving a real-world or
ESS01.03.04
mathematical problem; understand that a
CCR.EE.C
variable can represent an unknown number or,
depending on the purpose at hand, any
number in a specified set.
4.EE.7
Solve real-world and mathematical problems
CC.6.EE.7
by writing and solving equations of the form x
ESS01.03.05
+ p = q and px = q for cases in which p, q and
CCR.EE.C
x are all nonnegative rational numbers.
Write an inequality of the form x > c or x < c to CC.6.EE.8
4.EE.8
represent a constraint or condition in a realESS01.03.03
world or mathematical problem. Recognize
CCR.EE.C
that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of
such inequalities on number line diagrams.
Predicting pay on a
base wage as tips
Entering an
expression in a
spreadsheet
Projecting whether
certain budgets will
work
Keeping personal
finance records
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Represent and analyze quantitative relationships between dependent and independent
variables.
4.EE.9
Use variables to represent two quantities in a CC.6.EE.9
Reading
real-world problem that change in
ESS01.03.04
newspapers and
relationship to one another; write an equation CCR.EE.C
advertisements
to express one quantity, thought of as the
dependent variable, in terms of the other
quantity, thought of as the independent
variable. Analyze the relationship between
the dependent and independent variables
using graphs and tables, and relate these to
the equation. For example, in a problem
involving motion at constant speed, list and
graph ordered pairs of distances and times,
and write the equation d = 65t to represent
the relationship between distance and time.
Use properties of operations to generate equivalent expressions.
4.EE.10
Apply properties of operations as strategies
CC.7.EE.1
Finding a price
to add, subtract, factor, and expand linear
CCR.EE.D
increase of 10%
expressions with rational coefficients.
4.EE.11
Understand that rewriting an expression in
CC.7.EE.2
Entering an
different forms in a problem context can shed CCR.EE.D
expression in a
light on the problem and how the quantities in
spreadsheet
it are related. For example, a + 0.05a = 1.05a
means that “increase by 5%” is the same as
“multiply by 1.05.”
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Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
4.EE.12
Solve multi-step real-life and mathematical
CC.7.EE.3
Keeping personal
problems posed with positive and negative
ESS01.03.05
finance records
rational numbers in any form (whole
CCR.EE.D
numbers, fractions, and decimals), using
tools strategically. Apply properties of
operations as strategies to calculate with
numbers in any form; convert between forms
as appropriate; and assess the
reasonableness of answers using mental
computation and estimation strategies. For
example: If a woman making $25 an hour
gets a 10% raise, she will make an additional
1/10 of her salary an hour, or $2.50, for a
new salary of $27.50. If you want to place a
towel bar 9 3/4 inches long in the center of a
door that is 27 1/2 inches wide, you will need
to place the bar about 9 inches from each
edge; this estimate can be used as a check
on the exact computation.
4.EE.13
Use variables to represent quantities in a
CC.7.EE.4
Keeping personal
real-world or mathematical problem, and
ESS01.03.05
finance records or
construct simple equations and inequalities
CCR.EE.D
using a spreadsheet
to solve problems by reasoning about the
quantities.
For example, the
a. Solve word problems leading to
perimeter of a
equations of the form px + q = r
rectangle is 54 cm.
and p(x + q) = r, where p, q, and r
Its length is 6 cm.
are specific rational numbers.
What is its width?
Solve equations of these forms
fluently. Compare an algebraic
For example, as a
solution to an arithmetic solution,
salesperson, you
identifying the sequence of the
are paid $50 per
operations used in each approach.
week plus $3 per
b. Solve word problems leading to
sale. This week you
inequalities of the form px + q > r
want your pay to be
or px + q < r, where p, q, and r are
at least $100. Write
specific rational numbers. Graph
an inequality for the
the solution set of the inequality
number of sales you
and interpret it in the context of the
need to make, and
problem.
describe the
solutions.
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Work with radicals and integer exponents.
4.EE.14
Know and apply the properties of integer
exponents to generate equivalent numerical
expressions. For example, 32 × 3(–5) = 3(–3) =
1/(33) = 1/27.
4.EE.15
Use square root and cube root symbols to
represent solutions to equations of the form
x2 = p and x3 = p, where p is a positive
rational number. Evaluate square roots of
small perfect squares and cube roots of small
perfect cubes. Know that √2 is irrational.
4.EE.16
Use numbers expressed in the form of a
single digit times an integer power of 10 to
estimate very large or very small quantities,
and to express how many times as much one
is than the other.
CC.8.EE.1
CCR.EE.D
CC.8.EE.2
CCR.EE.D
CC.8.EE.3
CCR.EE.D
Estimating the
population of the
United States as 3 ×
108 and the
population of the
world as 7 × 109,
and determining that
the world population
is more than 20
times larger
4.EE.17
Perform operations with numbers expressed CC.8.EE.4
in scientific notation, including problems
CCR.EE.D
where both decimal and scientific notation
are used. Use scientific notation and choose
units of appropriate size for measurements of
very large or very small quantities (e.g., use
millimeters per year for seafloor spreading).
Interpret scientific notation that has been
generated by technology.
Understand the connections between proportional relationships, lines, and linear
equations.
4.EE.18
Graph proportional relationships, interpreting CC.8.EE.5
Comparing a
the unit rate as the slope of the graph.
ESS01.03.06
distance-time graph
Compare two different proportional
CCR.EE.D
to a distance-time
relationships represented in different ways.
equation to
determine which of
two moving objects
has greater speed
4.EE.19
Use similar triangles to explain why the slope CC.8.EE.6
Observing triangles
m is the same between any two distinct
in roof trusses
points on a non-vertical line in the coordinate
plane; derive the equation y = mx for a line
through the origin and the equation y = mx +
b for a line intercepting the vertical axis at b.
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Analyze and solve linear equations and pairs of simultaneous linear equations.
4.EE.20
Solve linear equations in one variable.
CC.8.EE.7
a. Give examples of linear equations in
CCR.EE.D
one variable with one solution,
infinitely many solutions, or no
solutions. Show which of these
possibilities is the case by
successively transforming the given
equation into simpler forms, until an
equivalent equation of the form x = a,
a = a, or a = b results (where a and b
are different numbers).
b. Solve linear equations with rational
number coefficients, including
equations whose solutions require
expanding expressions using the
distributive property and collecting like
terms.
4.EE.21
Analyze and solve pairs of simultaneous
CC.8.EE.8
linear equations.
CCR.EE.D
a. Understand that solutions to a system
of two linear equations in two
variables correspond to points of
intersection of their graphs, because
points of intersection satisfy both
equations simultaneously.
b. Solve systems of two linear equations
in two variables algebraically, and
estimate solutions by graphing the
equations. Solve simple cases by
inspection. For example, 3x + 2y = 5
and 3x + 2y = 6 have no solution
because 3x + 2y cannot
simultaneously be 5 and 6.
c. Solve real-world and mathematical
problems leading to two linear
equations in two variables. For
example, given coordinates for two
pairs of points, determine whether the
line through the first pair of points
intersects the line through the second
pair.
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FUNCTIONS (F)
Define, evaluate, and compare functions.
4.F.1
Understand that a function is a rule that
CC.8.F.1
assigns to each input exactly one output. The CCR.F.D
graph of a function is the set of ordered pairs
consisting of an input and the corresponding
output.
4.F.2
Compare properties of two functions each
CC.8.F.2
represented in a different way (algebraically, ESS01.03.06
graphically, numerically in tables, or by
CCR.F.IF.E
verbal descriptions). For example, given a
linear function represented by a table of
values and a linear function represented by
an algebraic expression, determine which
function has the greater rate of change.
Interpret the equation y = mx + b as defining CC.8.F.3
4.F.3
a linear function, whose graph is a straight
ESS01.03.06
line; give examples of functions that are not
CCR.F.D
2
linear. For example, the function A = s giving
the area of a square as a function of its side
length is not linear because its graph
contains the points (1,1), (2,4) and (3,9),
which are not on a straight line.
Use functions to model relationships between quantities.
4.F.4
Construct a function to model a linear
CC.8.F.4
relationship between two quantities.
ESS01.03.06
Determine the rate of change and initial value CCR.F.D
of the function from a description of a
relationship or from two (x, y) values,
including reading these from a table or from a
graph. Interpret the rate of change and initial
value of a linear function in terms of the
situation it models, and in terms of its graph
or a table of values.
4.F.5
Describe qualitatively the functional
CC.8.F.5
relationship between two quantities by
ESS01.03.06
analyzing a graph (e.g., where the function is CCR.F.D
increasing or decreasing, linear or nonlinear).
Sketch a graph that exhibits the qualitative
features of a function that has been
described verbally.
Reading a nutritional
graph on a health
poster
Conversing about
information
contained in graphs
in newspapers and
magazines
Reading a graph in
an ad or poster
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GEOMETRY (G)
Solve real-world and mathematical problems involving area, surface area, and volume.
4.G.1
Find area of right triangles, other triangles,
CC.6.G.1
Laying tile on a floor
special quadrilaterals, and polygons by
CCR.G.C
composing into rectangles or decomposing
into triangles and other shapes; apply these
techniques in the context of solving realworld and mathematical problems.
4.G.2
Find the volume of a right rectangular prism
CC.6.G.2
Filling a sandbox
with fractional edge lengths by packing it with
with sand or a
unit cubes of the appropriate unit fraction
garden with mulch
edge lengths, and show that the volume is
the same as would be found by multiplying
the edge lengths of the prism. Apply the
formulas V = l w h and V = b h to find
volumes of right rectangular prisms with
fractional edge lengths in the context of
solving real-world and mathematical
problems.
4.G.3
Draw polygons in the coordinate plane given CC.6.G.3
coordinates for the vertices; use coordinates CCR.G.C
to find the length of a side joining points with
the same first coordinate or the same second
coordinate. Apply these techniques in the
context of solving real-world and
mathematical problems.
4.G.4
Represent three-dimensional figures using
CC.6.G.4
Estimating the
nets made up of rectangles and triangles,
CCR.G.C
amount of fabric
and use the nets to find the surface area of
needed to cover a
these figures. Apply these techniques in the
piece of furniture
context of solving real-world and
mathematical problems.
Draw, construct, and describe geometrical figures and describe the relationships between
them.
4.G.5
Solve problems involving scale drawings of
CC.7.G.1
Buying carpet tiles
geometric figures, including computing actual CCR.G.D
or wallpaper
lengths and areas from a scale drawing and
reproducing a scale drawing at a different
scale.
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4.G.6
Draw (freehand, with ruler and protractor,
CC.7.G.2
and with technology) geometric shapes with
given conditions. Focus on constructing
triangles from three measures of angles or
sides, noticing when the conditions
determine a unique triangle, more than one
triangle, or no triangle.
4.G.7
Describe the two-dimensional figures that
CC.7.G.3
result from slicing three-dimensional figures,
as in plane sections of right rectangular
prisms and right rectangular pyramids.
Solve real-life and mathematical problems involving angle measure, area, surface area, and
volume.
4.G.8
Know the formulas for the area and
CC.7.G.4
circumference of a circle and use them to
CCR.G.D
solve problems; give an informal derivation of
the relationship between the circumference
and area of a circle.
4.G.9
Use facts about supplementary,
CC.7.G.5
Cutting molding at a
complementary, vertical, and adjacent angles CCR.G.D
correct angle so that
in a multi-step problem to write and solve
both ends meet with
simple equations for an unknown angle in a
no space between
figure.
4.G.10
Solve real-world and mathematical problems CC.7.G.6
Buying paint for a
involving area, volume and surface area of
ESS01.03.04
project
two- and three-dimensional objects
CCR.G.D
composed of triangles, quadrilaterals,
polygons, cubes, and right prisms.
Understand congruence and similarity using physical models, transparencies, or geometry
software.
4.G.11
Verify experimentally the properties of
CC.8.G.1
rotations, reflections, and translations:
a. Lines are taken to lines, and line
segments to line segments of the same
length.
b. Angles are taken to angles of the same
measure.
c. Parallel lines are taken to parallel lines.
4.G.12
Understand that a two-dimensional figure is
CC.8.G.2
congruent to another if the second can be
CCR.G.D
obtained from the first by a sequence of
rotations, reflections, and translations; given
two congruent figures, describe a sequence
that exhibits the congruence between them.
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4.G.13
Describe the effect of dilations, translations,
CC.8.G.3
rotations, and reflections on two-dimensional
figures using coordinates.
4.G.14
Understand that a two-dimensional figure is
CC.8.G.4
similar to another if the second can be
CCR.G.D
obtained from the first by a sequence of
rotations, reflections, translations, and
dilations; given two similar two-dimensional
figures, describe a sequence that exhibits the
similarity between them.
4.G.15
Use informal arguments to establish facts
CC.8.G.5
about the angle sum and exterior angle of
CCR.G.D
triangles, about the angles created when
parallel lines are cut by a transversal, and the
angle-angle criterion for similarity of triangles.
For example, arrange three copies of the
same triangle so that the three angles appear
to form a line, and give an argument in terms
of transversals why this is so.
Understand and apply the Pythagorean Theorem.
4.G.16
Explain a proof of the Pythagorean Theorem CC.8.G.6
and its converse.
4.G.17
Apply the Pythagorean Theorem to
CC.8.G.7
Calculating the
determine unknown side lengths in right
CCR.G.D
distance from roof
triangles in real-world and mathematical
peak to gutter from
problems in two and three dimensions.
the ground
4.G.18
Apply the Pythagorean Theorem to find the
CC.8.G.8
distance between two points in a coordinate
CCR.G.D
system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and
spheres.
4.G.19
Know the formulas for the volume of cones,
CC.8.G.9
cylinders, and spheres and use them to solve
real-world and mathematical problems.
STATISTICS AND PROBABILITY (SP)
Develop understanding of statistical variability.
4.SP.1
Recognize a statistical question as one that
CC.6.SP.1
anticipates variability in the data related to
CCR.SP.C
the question and accounts for it in the
answers. For example, “How old am I?” is not
a statistical question, but “How old are the
students in my class?” is a statistical
question because one anticipates variability
in students’ ages.
Understanding there
are no “true”
measures for
statistical quantities
(i.e., home values,
expected wages,
expected prices,
expected profits or
losses)
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4.SP.2
4.SP.3
Understand that a set of data collected to
answer a statistical question has a
distribution which can be described by its
center, spread, and overall shape.
Recognize that a measure of center for a
numerical data set summarizes all of its
values with a single number, while a
measure of variation describes how its
values vary with a single number.
CC.6.SP.2
CCR.SP.C
CC.6.SP.3
CCR.SP.C
Summarize and describe distributions.
4.SP.4
Display numerical data in plots on a number
CC.6.SP.4
line, including dot plots, histograms, and box ESS01.03.06
plots.
CCR.SP.C
4.SP.5
Summarize numerical data sets in relation to CC.6.SP.5
their context, such as by:
CCR.SP.D
a. Reporting the number of observations.
b. Describing the nature of the attribute
under investigation, including how it
was measured and its units of
measurement.
c. Giving quantitative measures of center
(median and/or mean) and variability
(interquartile range and/or mean
absolute deviation), as well as
describing any overall pattern and any
striking deviations from the overall
pattern with reference to the context in
which the data was gathered.
d. Relating the choice of measures of
center and variability to the shape of
the data distribution and the context in
which the data was gathered.
Use random sampling to draw inferences about a population.
4.SP.6
Understand that statistics can be used to
CC.7.SP.1
gain information about a population by
CCR.SP.D
examining a sample of the population;
generalizations about a population from a
sample are valid only if the sample is
representative of that population. Understand
that random sampling tends to produce
representative samples and support valid
inferences.
Taking political
action to institute
changes in the
community
Recording
information about
students in class
(i.e., age, number of
children, number of
years at current
residence)
Tracking one’s daily
expenses
Determining a grade
point average
Conducting a
political survey
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4.SP.7
Use data from a random sample to draw
inferences about a population with an
unknown characteristic of interest. Generate
multiple samples (or simulated samples) of
the same size to gauge the variation in
estimates or predictions.
CC.7.SP.2
CCR.SP.D
Draw informal comparative inferences about two populations.
4.SP.8
Informally assess the degree of visual
CC.7.SP.3
overlap of two numerical data distributions
CCR.SP.D
with similar variabilities, measuring the
difference between the centers by expressing
it as a multiple of a measure of variability. For
example, the mean height of players on the
basketball team is 10 cm greater than the
mean height of players on the soccer team,
about twice the variability (mean absolute
deviation) on either team; on a dot plot, the
separation between the two distributions of
heights is noticeable.
4.SP.9
Use measures of center and measures of
CC.7.SP.4
variability for numerical data from random
CCR.SP.D
samples to draw informal comparative
inferences about two populations.
Estimating the mean
word length in a
book by randomly
sampling words from
the book. Gauge
how far off the
estimate or
prediction might be.
Comparing personal
financial choices to
data from an ideal
spender’s purchase
log
Deciding whether
the words in a
chapter of NRS
Level 1 science
book are generally
longer than the
words in a chapter
of NRS level 4
science book
Investigate chance processes and develop, use, and evaluate probability models.
4.SP.10
Understand that the probability of a chance
CC.7.SP.5
Deciding whether or
event is a number between 0 and 1 that
CCR.SP.D
not to carry an
expresses the likelihood of the event
umbrella
occurring. Larger numbers indicate greater
likelihood. A probability near 0 indicates an
unlikely event, a probability around 1/2
indicates an event that is neither unlikely nor
likely, and a probability near 1 indicates a
likely event.
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4.SP.11
Approximate the probability of a chance
event by collecting data on the chance
process that produces it and observing its
long-run relative frequency, and predict the
approximate relative frequency given the
probability.
CC.7.SP.6
CCR.SP.D
4.SP.12
Develop a probability model and use it to find
probabilities of events. Compare probabilities
from a model to observed frequencies; if the
agreement is not good, explain possible
sources of the discrepancy.
a. Develop a uniform probability model
by assigning equal probability to all
outcomes, and use the model to
determine probabilities of events.
b. Develop a probability model (which
may not be uniform) by observing
frequencies in data generated from a
chance process.
CC.7.SP.7
CCR.SP.D
When rolling a
number cube 600
times, predicting that
a 3 or 6 would be
rolled roughly 200
times, but probably
not exactly 200
times.
Tossing a coin
Selecting a student
at random from a
class, finding the
probability that Jane
will be selected and
the probability that
another female
student will be
selected.
Finding the
approximate
probability that a
spinning penny will
land heads up or
that a tossed paper
cup will land openend down. Do the
outcomes for the
spinning penny
appear to be equally
likely based on the
observed
frequencies?
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4.SP.13
Find probabilities of compound events using
CC.7.SP.8
organized lists, tables, tree diagrams, and
CCR.SP.D
simulation.
a. Understand that, just as with simple
events, the probability of a compound
event is the fraction of outcomes in the
sample space for which the compound
event occurs.
b. Represent sample spaces for
compound events using methods such
as organized lists, tables and tree
diagrams.
c. Design and use a simulation to
generate frequencies for compound
events.
Investigate patterns of association in bivariate data.
4.SP.14
Construct and interpret scatter plots for
bivariate measurement data to investigate
patterns of association between two
quantities. Describe patterns such as
clustering, outliers, positive or negative
association, linear association, and nonlinear
association.
4.SP.15
Know that straight lines are widely used to
model relationships between two quantitative
variables. For scatter plots that suggest a
linear association, informally fit a straight
line, and informally assess the model fit by
judging the closeness of the data points to
the line.
CC.8.SP.1
CCR.SP.D
CC.8.SP.2
CCR.SP.D
Rolling dice or
tossing a coin
For an event
described in
everyday language
(e.g., “rolling double
sixes”), identifying
the outcomes in the
sample space which
compose the event
Use random digits
as a simulation tool
to approximate the
answer to the
question. If 40% of
donors have type A
blood, what is the
probability that it will
take at least 4
donors to find one
with type A blood?
Plot and analyze
related student
attributes (i.e., bed
times and wake up
times, workdays per
week and times
eating out per week,
number of children,
number of hours of
sleep per night)
Plot and analyze
related student
attributes (i.e., bed
times and wake up
times, workdays per
week and times
eating out per week,
number of children,
number of hours of
sleep per night)
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4.SP.16
Use the equation of a linear model to solve
problems in the context of bivariate
measurement data, interpreting the slope
and intercept. For example, in a linear model
for a biology experiment, interpret a slope of
1.5 cm/hr as meaning that an additional hour
of sunlight each day is associated with an
additional 1.5 cm in mature plant height.
CC.8.SP.3
CCR.SP.D
4.SP.17
Understand that patterns of association can
CC.8.SP.4
also be seen in bivariate categorical data by
ESS01.03.06
displaying frequencies and relative
CCR.SP.D
frequencies in a two-way table. Construct
and interpret a two-way table summarizing
data on two categorical variables collected
from the same subjects. Use relative
frequencies calculated for rows or columns to
describe possible association between the
two variables.
Plot and analyze
related student
attributes (i.e., bed
times and wake up
times, workdays per
week and times
eating out per week,
number of children,
number of hours of
sleep per night)
Collect data from
students on whether
or not they drink
coffee and whether
or not they have
trouble sleeping at
night. Is there
evidence that those
who drink coffee
also tend to have
trouble sleeping at
night?
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NRS Level 5 Overview
Number and Quantity (N)
The Real Number System (RN)
 Extend the properties of exponents to rational exponents
 Use properties of rational and irrational numbers
Quantities (Q)
 Reason quantitatively and use units to solve problems
Algebra (A)
Seeing Structure in Expressions (SSE)
 Interpret the structure of expressions
 Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Expressions (APR)
 Perform arithmetic operations on polynomials
Creating Equations (CED)
 Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities (REI)
 Understand solving equations as a process of reasoning and explain the
reasoning
 Solve equations and inequalities in one variable
 Solve systems of equations
 Represent and solve equations and inequalities graphically
Functions (F)
Interpreting Functions (IF)
 Analyze functions using different representations
Building Functions (BF)
 Build a function that models a relationship between two quantities
Linear, Quadratic, and Exponential Models (LE)
 Construct and compare linear, quadratic, and exponential models and solve
problems
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NRS Level 5 Overview (continued)
Geometry (G)
Congruence (CO)
 Experiment with transformations in the plane
 Understand congruence in terms of rigid motions
 Prove geometric theorems
 Make geometric constructions
Similarity, Right Triangles, and Trigonometry (SRT)
 Understand similarity in terms of similarity transformations
 Prove theorems involving similarity
Circles (C)
 Understand and apply theorems about circles
 Find arc lengths and areas of sectors
Expressing Geometric Properties with Equations (GPE)
 Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension (GMD)
 Explain volume formulas and use them to solve problems
Modeling with Geometry (MG)
 Apply geometric concepts in modeling situations
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NRS Level 5 Overview (continued)
Statistics and Probability (S)
Interpreting Categorical and Quantitative Data (ID)
 Summarize, represent, and interpret data on a
single count or measurement variable
 Summarize, represent, and interpret data on
two categorical and quantitative variables
 Interpret linear models
Making Inferences and Justifying Conclusions (IC)
 Understand and evaluate random processes
underlying statistical experiments
 Make inferences and justify conclusions from
sample surveys, experiments, and
observational studies of circles
Using Probability to Make Decisions (MD)
 Calculate expected values and use them to
solve problems
 Use probability to evaluate outcomes of
decisions
★
+
Modeling
Provides foundations for advanced college courses
Mathematical Practices
1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
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OVERVIEW EXPLANATION OF MATHEMATICS
NRS LEVELS 5 & 6 – Low & High Adult Secondary Education
NRS Level 5 (Grade Levels 9.0-10.9)
NRS Level 6 (Grade Levels 11.0-12.9)
Difference between Adult Education and High School Education
The literacy through low intermediate standards presented in this document provide for
a fairly sequential progression of mathematics instruction and learning. Low and high
ASE level standards together prepare adult education students for the rigors of high
school level mathematics.
The low and high ASE level standards are grouped into six conceptual categories, each
of which is further divided into domain groupings. Adult educators should be aware that
specific order, as with all standards, should be left up to individual teachers and
programs, not necessarily having to complete all of Level 5 before moving on to Level 6.
College and Career Ready
Low and high ASE standards specify the mathematics that all students should study in
order to be college and career ready20. Additional mathematics that students should
learn in order to take advanced courses such as calculus, advanced statistics, or
discrete mathematics are indicated by (+), as in this example:
(+) Represent complex numbers on the complex plane in rectangular
and polar form (including real and imaginary numbers).
All standards without a (+) symbol should be in the common mathematics curriculum for
all college and career ready students. Standards with a (+) symbol may also appear in
courses intended for all students.
It should also be noted that the goal is to be college and career ready, and levels 5 and
6 represent the entirety of this goal. Level 5 is not exclusively for GED/Career
preparedness, and Level 6 is not exclusively college ready.
Notes on Courses and Transitions
Currently, most high schools use a traditional sequence of Algebra I, Geometry, and
Algebra II. This has been slowly changing over the years, as some schools see the
advantage of using an Algebra I, Algebra II, and then Geometry sequence. These
standards do not mandate a sequence of courses.
20
Massachusetts Curriculum Framework for Mathematics, 2011, www.doe.mass.edu/frameworks/current
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A major transition for adults is the transition from adult education to post-secondary
education for college and careers. The evidence concerning college and career
readiness shows clearly that the knowledge, skills, and practices important for
readiness include a great deal of mathematics. Important standards for college and
career readiness are distributed across levels and courses. Some of the highest priority
content for college and career readiness comes from NRS level 4 (5.0-8.9). This body of
material includes powerfully useful proficiencies such as applying ratio reasoning in
real-world and mathematical problems, computing fluently with positive and negative
fractions and decimals, and solving real-world and mathematical problems involving
angle measure, area, surface area, and volume. It is important to note as well that
information generated by assessment systems for college and career readiness should
be developed in collaboration with representatives from higher education and workforce
development programs, and should be validated by subsequent performance of
students in college and the workforce.
Explanation of Conceptual Categories
The Adult Secondary Education standards are listed in conceptual categories:
I.
Number and Quantity
II.
Algebra
III.
Functions
IV.
Modeling
V.
Geometry
VI.
Statistics and Probability
These six conceptual categories portray a coherent view of Adult Secondary Education
based on the common core high school standards; a student’s work with functions, for
example, crosses a number of traditional course boundaries, potentially up through and
including calculus.
Conceptual Category I - Number and Quantity
Numbers and Number Systems - During NRS Levels 1-4, adult education students
must repeatedly extend their conception of number. At first, “number” means “counting
number”: 1, 2, 3... Soon after that, 0 is used to represent “none” and the whole numbers
are formed by the counting numbers together with zero. The next extension is fractions.
At first, fractions are barely numbers and tied strongly to pictorial representations. Yet
by the time students understand division of fractions, they have a strong concept of
fractions as numbers and have connected them, via their decimal representations, with
the base-ten system used to represent the whole numbers. Also, during levels 1-4,
fractions are augmented by negative fractions to form the rational numbers. And during
NRS Level 4 students extend this system once more, augmenting the rational numbers
with the irrational numbers to form the real numbers. ASE level students will be exposed
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to yet another extension of number, when the real numbers are augmented by the
imaginary numbers to form the complex numbers.
With each extension of number, the meanings of addition, subtraction, multiplication,
and division are extended. In each new number system—integers, rational numbers,
real numbers, and complex numbers—the four operations stay the same in two
important ways: They have the commutative, associative, and distributive properties,
and their new meanings are consistent with their previous meanings.
Quantities - In real world problems, the answers are usually not numbers but
quantities: numbers with units, which involves measurement. ASE students
encounter a wider variety of units in modeling (e.g., acceleration, currency
conversions, derived quantities such as person-hours and heating degree days,
social science rates such as per-capita income, and rates in everyday life such as
points scored per game or batting averages). They also encounter novel situations
in which they themselves must conceive the attributes of interest. For example, to
find a good measure of overall highway safety, they might propose measures such
as fatalities per year, fatalities per year per driver, or fatalities per vehicle-mile
traveled. Such a conceptual process is sometimes called quantification.
Conceptual Category II – Algebra
Expressions - An expression is a record of a computation with numbers, symbols that
represent numbers, arithmetic operations, exponentiation, and, at more advanced
levels, the operation of evaluating a function.
Reading an expression with comprehension involves analysis of its underlying structure.
This may suggest a different but equivalent way of writing the expression that exhibits
some different aspect of its meaning. For example, p + 0.05p can be interpreted as the
addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax
is the same as multiplying the price by a constant factor.
Algebraic manipulations are governed by the properties of operations and exponents
and the conventions of algebraic notation. At times, an expression is the result of
applying operations to simpler expressions. For example, p + 0.05p is the sum of the
simpler expressions p and 0.05p. Viewing an expression as the result of operation on
simpler expressions can sometimes clarify its underlying structure.
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A spreadsheet or a computer algebra system (CAS) can be used to experiment with
algebraic expressions, perform complicated algebraic manipulations, and understand
how algebraic manipulations behave.
Equations and Inequalities - An equation is a statement of equality between two
expressions, often viewed as a question asking for which values of the variables the
expressions on either side are in fact equal. These values are the solutions to the
equation. An identity, in contrast, is true for all values of the variables; identities are
often developed by rewriting an expression in an equivalent form.
The same solution techniques used to solve equations can be used to rearrange
formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be
solved for h using the same deductive process.
Inequalities can be solved by reasoning about the properties of inequality. Many, but not
all, of the properties of equality continue to hold for inequalities and can be useful in
solving them.
Connections to Functions and Modeling - Expressions can define functions, and
equivalent expressions define the same function. Asking when two functions have the
same value for the same input leads to an equation; graphing the two functions allows
for finding approximate solutions of the equation. Converting a verbal description to an
equation, inequality, or system of these is an essential skill in modeling.
Conceptual Category III – Functions
Functions describe situations where one quantity determines another. For example, the
return on $10,000 invested at an annualized percentage rate of 4.25% is a function of
the length of time the money is invested. Because we continually make theories about
dependencies between quantities in nature and society, functions are important tools in
the construction of mathematical models.
In mathematics, functions usually have numerical inputs and outputs and are often
defined by an algebraic expression. For example, the time in hours it takes for a car to
drive 100 miles is a function of the car’s speed in miles per hour, v; the rule T(v) = 100/v
expresses this relationship algebraically and defines a function whose name is T.
Functions presented as expressions can model many important phenomena. Two
important families of functions characterized by laws of growth are linear functions,
which grow at a constant rate, and exponential functions, which grow at a constant
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percent rate. Linear functions with a constant term of zero describe proportional
relationships.
A graphing utility or a computer algebra system can be used to experiment with
properties of these functions and their graphs and to build computational models of
functions, including recursively defined functions.
Connections to Expressions, Equations, Modeling, and Coordinates - Determining
an output value for a particular input involves evaluating an expression; finding inputs
that yield a given output involves solving an equation. Questions about when two
functions have the same value for the same input lead to equations, whose solutions
can be visualized from the intersection of their graphs. Because functions describe
relationships between quantities, they are frequently used in modeling. Sometimes
functions are defined by a recursive process, which can be displayed effectively using a
spreadsheet or other technology.
Conceptual Category IV – Modeling
Modeling links classroom mathematics and statistics to everyday life, work, and
decision-making. Modeling is the process of choosing and using appropriate
mathematics and statistics to analyze empirical situations, to understand them better,
and to improve decisions. Quantities and their relationships in physical, economic,
public policy, social, and everyday situations can be modeled using mathematical and
statistical methods. When making mathematical models, technology is valuable for
varying assumptions, exploring consequences, and comparing predictions with data.
A model can be very simple, such as writing total cost as a product of unit price and
number bought, or using a geometric shape to describe a physical object like a coin.
Even such simple models involve making choices. It is up to us whether to model a coin
as a three-dimensional cylinder, or whether a two-dimensional disk works well enough
for our purposes. Other situations—modeling a delivery route, a production schedule, or
a comparison of loan amortizations—need more elaborate models that use other tools
from the mathematical sciences. Real-world situations are not organized and labeled for
analysis; formulating tractable models, representing such models, and analyzing them is
appropriately a creative process. Like every such process, this depends on acquired
expertise as well as creativity.
One of the insights provided by mathematical modeling is that essentially the same
mathematical or statistical structure can sometimes model seemingly different
situations. Models can also shed light on the mathematical structures themselves, for
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example, as when a model of bacterial growth makes more vivid the explosive growth of
the exponential function.
The basic modeling cycle is summarized in the above diagram. It involves the following:
1. Identifying variables in the situation and selecting those that represent essential
features.
2. Formulating a model by creating and selecting geometric, graphical, tabular,
algebraic, or statistical representations that describe relationships between the
variables.
3. Analyzing and performing operations on these relationships to draw conclusions.
4. Interpreting the results of the mathematics in terms of the original situation.
5. Validating the conclusions by comparing them with the situation, and then either
improving the model or, if it is acceptable,
6. Reporting on the conclusions and the reasoning behind them. Choices,
assumptions, and approximations are present throughout this cycle.
In descriptive modeling, a model simply describes the phenomena or summarizes them
in a compact form. Graphs of observations are a familiar descriptive model—for
example, graphs of global temperature and atmospheric CO2 over time.
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit
with parameters that are empirically based; for example, exponential growth of bacterial
colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from
a constant reproduction rate. Functions are an important tool for analyzing such
problems.
Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry
software are powerful tools that can be used to model purely mathematical phenomena
(e.g., the behavior of polynomials) as well as physical phenomena.
Modeling is best interpreted not as a collection of isolated topics but rather in relation to
other standards. Making mathematical models is a Standard for Mathematical Practice,
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and specific modeling standards appear throughout the ASE standards indicated by a
star symbol (★).
Conceptual Category V – Geometry
An understanding of the attributes and relationships of geometric objects can be applied
in diverse contexts—interpreting a schematic drawing, estimating the amount of wood
needed to frame a sloping roof, rendering computer graphics, or designing a sewing
pattern for the most efficient use of material.
Although there are many types of geometry, school mathematics is devoted primarily to
plane Euclidean geometry, studied both synthetically (without coordinates) and
analytically (with coordinates). Euclidean geometry is characterized most importantly by
the Parallel Postulate, that through a point not on a given line there is exactly one
parallel line. (Spherical geometry, in contrast, has no parallel lines.)
The concepts of congruence, similarity, and symmetry can be understood from the
perspective of geometric transformation. Fundamental are the rigid motions:
translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally). Reflections
and rotations each explain a particular type of symmetry, and the symmetries of an
object offer insight into its attributes—as when the reflective symmetry of an isosceles
triangle assures that its base angles are congruent.
Analytic geometry connects algebra and geometry, resulting in powerful methods of
analysis and problem solving. Just as the number line associates numbers with
locations in one dimension, a pair of perpendicular axes associates pairs of numbers
with locations in two dimensions. This correspondence between numerical coordinates
and geometric points allows methods from algebra to be applied to geometry and vice
versa. The solution set of an equation becomes a geometric curve, making visualization
a tool for doing and understanding algebra. Geometric shapes can be described by
equations, making algebraic manipulation into a tool for geometric understanding,
modeling, and proof. Geometric transformations of the graphs of equations correspond
to algebraic changes in their equations.
Dynamic geometry environments provide students with experimental and modeling tools
that allow them to investigate geometric phenomena in much the same way as
computer algebra systems allow them to experiment with algebraic phenomena.
Connections to Equations - The correspondence between numerical coordinates and
geometric points allows methods from algebra to be applied to geometry and vice versa.
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The solution set of an equation becomes a geometric curve, making visualization a tool
for doing and understanding algebra. Geometric shapes can be described by equations,
making algebraic manipulation into a tool for geometric understanding, modeling, and
proof.
Conceptual Category VI – Statistics and Probability★
Decisions or predictions are often based on data—numbers in context. These decisions
or predictions would be easy if the data always sent a clear message, but the message
is often obscured by variability. Statistics provides tools for describing variability in data
and for making informed decisions that take it into account.
Data are gathered, displayed, summarized, examined, and interpreted to discover
patterns and deviations from patterns. Quantitative data can be described in terms of
key characteristics: measures of shape, center, and spread. The shape of a data
distribution might be described as symmetric, skewed, flat, or bell shaped, and it might
be summarized by a statistic measuring center (such as mean or median) and a statistic
measuring spread (such as standard deviation or interquartile range). Different
distributions can be compared numerically using these statistics or compared visually
using plots. Knowledge of center and spread are not enough to describe a distribution.
Which statistics to compare, which plots to use, and what the results of a comparison
might mean, depend on the question to be investigated and the real-life actions to be
taken.
Randomization has two important uses in drawing statistical conclusions. First,
collecting data from a random sample of a population makes it possible to draw valid
conclusions about the whole population, taking variability into account. Second,
randomly assigning individuals to different treatments allows a fair comparison of the
effectiveness of those treatments. A statistically significant outcome is one that is
unlikely to be due to chance alone, and this can be evaluated only under the condition
of randomness. The conditions under which data are collected are important in drawing
conclusions from the data; in critically reviewing uses of statistics in public media and
other reports, it is important to consider the study design, how the data were gathered,
and the analyses employed, as well as the data summaries and the conclusions drawn.
Random processes can be described mathematically by using a probability model: a list
or description of the possible outcomes (the sample space), each of which is assigned a
probability. In situations such as flipping a coin, rolling a number cube, or drawing a
card, it might be reasonable to assume various outcomes are equally likely. In a
probability model, sample points represent outcomes and combine to make up events;
probabilities of events can be computed by applying the Addition and Multiplication
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Rules. Interpreting these probabilities relies on an understanding of independence and
conditional probability, which can be approached through the analysis of two-way
tables.
Technology plays an important role in statistics and probability by making it possible to
generate plots, regression functions, and correlation coefficients, and to simulate many
possible outcomes in a short amount of time.
Connections to Functions and Modeling - Functions may be used to describe data; if
the data suggest a linear relationship, the relationship can be modeled with a regression
line, and its strength and direction can be expressed through a correlation coefficient.
The Standards for Mathematical Practice complement the content standards so that
students increasingly engage with the subject matter as they grow in mathematical
understanding and expertise.
Mathematical Practices
1.
Make sense of problems and
persevere in solving them.
★
+
Modeling
Provides foundations for advanced college courses
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
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NRS Level 5 – Low Adult Secondary Education
(Grade Levels 9.0-10.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This Skill
NUMBER AND QUANTITY (N)
The Real Number System (RN)
Extend the properties of exponents to rational exponents.
5.N.RN.1
Explain how the definition of the meaning CC.N.RN.1
of rational exponents follows from
extending the properties of integer
exponents to those values, allowing for a
notation for radicals in terms of rational
exponents. For example, we define 51/3 to
be the cube root of 5 because we want
3
3
3
(51/3 ) = 5(1/3) to hold, so (51/3 ) must
equal 5.
5.N.RN.2
Rewrite expressions involving radicals
CC.N.RN.2
and rational exponents using the
CCR.N.RN.E
properties of exponents.
Use properties of rational and irrational numbers.
5.N.RN.3
Explain why the sum or product of two
CC.N.RN.3
rational numbers is rational; that the sum
of a rational number and an irrational
number is irrational; and that the product
of a nonzero rational number and an
irrational number is irrational.
QUANTITIES★ (Q)
Reason quantitatively and use units to solve problems.
5.N.Q.1
Use units as a way to understand
CC.N.Q.1
problems and to guide the solution of
CCR.N.Q.E
multi-step problems; choose and interpret
units consistently in formulas; choose and
interpret the scale and the origin in graphs
and data displays.
Conversing about
information contained
in newspapers and
magazines
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ALGEBRA (A)
Seeing Structure in Expressions (SSE)
Interpret the structure of expressions.
5.A.SSE.1 Interpret expressions that represent a
CC.A.SSE.1
CCR.A.SSE.E
quantity in terms of its context.★
a. Interpret parts of an expression,
such as terms, factors, and
coefficients.
b. Interpret complicated expressions
by viewing one or more of their
parts as a single entity. For
example, interpret P(1 + r)n as the
product of P and a factor not
depending on P.
5.A.SSE.2 Use the structure of an expression to
CC.A.SEE.2
identify ways to rewrite it. For example,
CCR.A.SSE.E
4
4
2 )2
2
2
see x − y as(x
− (y ) , thus
recognizing it as a difference of squares
that can be factored as
(x 2 − y 2 )(x 2 + y 2 ).
Write expressions in equivalent forms to solve problems.
5.A.SSE.3 Choose and produce an equivalent form
CC.A.SSE.3
of an expression to reveal and explain
CCR.A.SSE.E
properties of the quantity represented by
the expression.★
a. Factor a quadratic expression to
reveal the zeros of the function it
defines.
Entering an
expression in a
spreadsheet
Keeping personal
finance records
Understanding
exponential growth
of bacteria or virus
such as HIV
ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS (APR)
Perform arithmetic operations on polynomials.
5.A.APR.1 Understand that polynomials form a
system analogous to the integers, namely,
they are closed under the operations of
addition, subtraction, and multiplication;
add, subtract, and multiply polynomials.
CC.A.APR.1
ESS01.03.04
ESS01.03.05
CCR.A.APR.E
Buying fencing to
expand an existing
garden
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CREATING EQUATIONS (CED)
Create equations that describe numbers or relationships.
5.A.CED.1 Create equations and inequalities in one
CC.A.CED.1
variable and use them to solve problems. CCR.A.CED.E
Include equations arising from linear and
quadratic functions, and simple rational
and exponential functions.
5.A.CED.2 Create equations in two or more variables CC.A.CED.2
to represent relationships between
CCR.A.CED.E
quantities; graph equations on coordinate
axes with labels and scales.
5.A.CED.3 Represent constraints by equations or
inequalities, and by systems of equations
and/or inequalities, and interpret solutions
as viable or nonviable options in a
modeling context.
CC.A.CED.3
CCR.A.CED.E
5.A.CED.4 Rearrange formulas to highlight a quantity
of interest, using the same reasoning as
in solving equations.
CC.A.CED.4
CCR.A.CED.E
Plot and analyze
related student
attributes (i.e.,
examples of quadratic
distributions needed)
Scatter plot and
analyze related
student attributes
(i.e., examples of
exponential and
polynomial
distributions needed)
Representing
inequalities
describing nutritional
and cost constraints
on combinations of
different foods
Rearranging Ohm’s
law V = IR to highlight
resistance R
REASONING WITH EQUATIONS AND INEQUALITIES (REI)
Understand solving equations as a process of reasoning and explain the reasoning.
5.A.REI.1 Explain each step in solving a simple
CC.A.REI.1
equation as following from the equality of
CCR.A.REI.E
numbers asserted at the previous step,
starting from the assumption that the
original equation has a solution. Construct
a viable argument to justify a solution
method.
Solve equations and inequalities in one variable.
5.A.REI.2 Solve linear equations and inequalities in
CC.A.REI.3
Finding an unknown
one variable, including equations with
CCR.A.REI.E amount of a check
coefficients represented by letters.
given current balance
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5.A.REI.3
Solve quadratic equations in one variable. CC.A.REI.4
a. Use the method of completing the
CCR.A.REI.E
square to transform any quadratic
equation in x into an equation of
the form (𝑥 − 𝑝)2 = 𝑞 that has the
same solutions. Derive the
quadratic formula from this form.
b. Solve quadratic equations by
inspection (𝑒. 𝑔. , 𝑓𝑜𝑟 𝑥 2 = 49),
taking square roots, completing the
square, the quadratic formula and
factoring, as appropriate to the
initial form of the equation.
Recognize when the quadratic
formula gives complex solutions
and write them as a ± bi for real
numbers a and b.
Solve systems of equations.
5.A.REI.4 Prove that, given a system of two
CC.A.REI.5
equations in two variables, replacing one
equation by the sum of that equation and
a multiple of the other produces a system
with the same solutions.
5.A.REI.5 Solve systems of linear equations exactly CC.A.REI.6
and approximately (e.g., with graphs),
CCR.A.REI.E
focusing on pairs of linear equations in
two variables.
Represent and solve equations and inequalities graphically.
5.A.REI.6 Understand that the graph of an equation CC.A.REI.10
in two variables is the set of all its
CCR.A.REI.E
solutions plotted in the coordinate plane,
often forming a curve (which could be a
line).
Plot and analyze
related student
attributes (i.e.,
examples of quadratic
distributions needed)
Comparing cost of
production to profit for
a bake sale
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FUNCTIONS (F)
INTERPRETING FUNCTIONS (IF)
Analyze functions using different representations.
5.F.IF.1
Graph functions expressed symbolically
and show key features of the graph, by
hand in simple cases and using
technology for more complicated cases.★
5.F.IF.2
Graph linear and quadratic functions and
show intercepts, maxima, and minima.
CC.F.IF.7
CCR.F.IF.E
Scatter plot and
analyze related
student attributes
(i.e., examples of
exponential and
polynomial
distributions needed)
CC.F.IF.7a
BUILDING FUNCTIONS (BF)
Build a function that models a relationship between two quantities.
5.F.BF.1
Write a function that describes a
CC.F.BF.1
relationship between two quantities.
CCR.F.BF.E
5.F.BF.2
Determine an explicit expression, a
recursive process, or steps for calculation
from a context.
Figuring the effect on
mortgage payments
of a change in
interest rates
CC.F.BF.2
LINEAR, QUADRATIC, AND EXPONENTIAL MODELS★ (LE)
Construct and compare linear, quadratic, and exponential models and solve problems.
5.F.LE.1
Construct linear and exponential
CC.F.LE.2
Setting realistic
functions, including arithmetic and
investment goals (i.e.,
geometric sequences, given a graph, a
college or retirement)
description of a relationship, or two inputoutput pairs (including reading these from
a table).
GEOMETRY (G)
CONGRUENCE (CO)
Experiment with transformations in the plane.
5.G.CO.1 Know precise definitions of angle, circle,
perpendicular line, parallel line, and line
segment, based on the undefined notions
of point, line, distance along a line, and
distance around a circular arc.
CC.G.CO.1
CCR.G.CO.E
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5.G.CO.2
Represent transformations in the plane
using transparencies and geometry
software; describe transformations as
functions that take points in the plane as
inputs and give other points as outputs.
Compare transformations that preserve
distance and angle to those that do not
(e.g., translation versus horizontal
stretch).
5.G.CO.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto
itself.
5.G.CO.4 Develop definitions of rotations,
reflections, and translations in terms of
angles, circles, perpendicular lines,
parallel lines, and line segments.
5.G.CO.5 Given a geometric figure and a rotation,
reflection, or translation, draw the
transformed figure using graph paper,
tracing paper, or geometry software.
Specify a sequence of transformations
that will carry a given figure onto another.
Understand congruence in terms of rigid motions.
5.G.CO.6 Use geometric descriptions of rigid
motions to transform figures and predict
the effect of a given rigid motion on a
given figure; given two figures, use the
definition of congruence in terms of rigid
motions to decide if they are congruent.
5.G.CO.7 Use the definition of congruence in terms
of rigid motions to show that two triangles
are congruent if and only if corresponding
pairs of sides and corresponding pairs of
angles are congruent.
5.G.CO.8 Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms
of rigid motions.
CC.G.CO.2
CC.G.CO.3
CC.G.CO.4
CC.G.CO.5
CC.G.CO.6
CC.G.CO.7
CC.G.CO.8
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Prove geometric theorems.
5.G.CO.9 Prove theorems about lines and angles.
Theorems include: vertical angles are
congruent; when a transversal crosses
parallel lines, alternate interior angles are
congruent and corresponding angles are
congruent; points on a perpendicular
bisector of a line segment are exactly
those equidistant from the segment’s
endpoints.
5.G.CO.10 Prove theorems about triangles.
Theorems include: measures of interior
angles of a triangle sum to 180°; base
angles of isosceles triangles are
congruent; the segment joining midpoints
of two sides of a triangle is parallel to the
third side and half the length; the medians
of a triangle meet at a point.
5.G.CO.11 Prove theorems about parallelograms.
Theorems include: opposite sides are
congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect
each other, and conversely, rectangles
are parallelograms with congruent
diagonals.
Make geometric constructions.
5.G.CO.12 Make formal geometric constructions with
a variety of tools and methods (compass
and straightedge, string, reflective
devices, paper folding, dynamic geometric
software, etc.). Copying a segment;
copying an angle; bisecting a segment;
bisecting an angle; constructing
perpendicular lines, including the
perpendicular bisector of a line segment;
and constructing a line parallel to a given
line through a point not on the line.
5.G.CO.13 Construct an equilateral triangle, a
square, and a regular hexagon inscribed
in a circle.
CC.G.CO.9
CC.G.CO.10
CC.G.CO.11
CC.G.CO.12
CC.G.CO.13
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Similarity, Right Triangles, and Trigonometry (SRT)
Understand similarity in terms of similarity transformations.
5.G.SRT.1 Verify experimentally the properties of
CC.G.SRT.1
dilations given by a center and a scale
factor:
a. A dilation takes a line not passing
through the center of the dilation to
a parallel line, and leaves a line
passing through the center
unchanged.
b. The dilation of a line segment is
longer or shorter in the ratio given
by the scale factor.
5.G.SRT.2 Given two figures, use the definition of
CC.G.SRT.2
similarity in terms of similarity
transformations to decide if they are
similar; explain using similarity
transformations the meaning of similarity
for triangles as the equality of all
corresponding pairs of angles and the
proportionality of all corresponding pairs
of sides.
5.G.SRT.3 Use the properties of similarity
CC.G.SRT.3
transformations to establish the AA
criterion for two triangles to be similar.
Prove theorems involving similarity.
5.G.SRT.4 Prove theorems about triangles.
CC.G.SRT.4
Theorems include: a line parallel to one
side of a triangle divides the other two
proportionally; and conversely, the
Pythagorean Theorem proved using
triangle similarity.
5.G.SRT.5 Use congruence and similarity criteria for
CC.G.SRT.5
triangles to solve problems and to prove
CCR.G.SRT.E
relationships in geometric figures.
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CIRCLES (C)
Understand and apply theorems about circles.
5.G.C.1
Prove that all circles are similar.
5.G.C.2
Identify and describe relationships among
inscribed angles, radii, and chords.
Include the relationship between central,
inscribed, and circumscribed angles;
inscribed angles on a diameter are right
angles; the radius of a circle is
perpendicular to the tangent where the
radius intersects the circle.
5.G.C.3
Construct the inscribed and circumscribed
circles of a triangle, and prove properties
of angles for a quadrilateral inscribed in a
circle.
5.G.C.4
(+) Construct a tangent line from a point
outside a given circle to the circle.
Find arc lengths and areas of sectors of circles.
5.G.C.5
Derive using similarity the fact that the
length of the arc intercepted by an angle
is proportional to the radius, and define
the radian measure of the angle as the
constant of proportionality; derive the
formula for the area of a sector.
CC.G.C.1
CC.G.C.2
CC.G.C.3
CC.G.C.4
CC.G.C.5
EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS (GPE)
Use coordinates to prove simple geometric theorems algebraically.
5.G.GPE.1 Prove the slope criteria for parallel and
CC.G.GPE.5
perpendicular lines and use them to
solve geometric problems (e.g., find the
equation of a line parallel or
perpendicular to a given line that passes
through a given point).
5.G.GPE.2 Find the point on a directed line segment CC.G.GPE.6
between two given points that partitions
the segment in a given ratio.
5.G.GPE.3 Use coordinates to compute perimeters
CC.G.GPE.7
of polygons and areas of triangles and
rectangles (e.g., using the distance
formula).★
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GEOMETRIC MEASUREMENT AND DIMENSION (GMD)
Explain volume formulas and use them to solve problems.
5.G.GMD.1 Give an informal argument for the
CC.G.GMD.1
formulas for the circumference of a
circle, area of a circle, volume of a
cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s
principle, and informal limit arguments.
5.G.GMD.2 Use volume formulas for cylinders,
CC.G.GMD.3
pyramids, cones, and spheres to solve
CCR.G.GMD.E
problems.★
MODELING WITH GEOMETRY (MG)
Apply geometric concepts in modeling situations.
5.G.MG.1
Use geometric shapes, their measures,
and their properties to describe objects
(e.g., modeling a tree trunk or a human
torso as a cylinder).★
5.G.MG.2
Apply concepts of density based on
area and volume in modeling situations
(e.g., persons per square mile, BTUs
per cubic foot).★
CC.G.MG.1
CC.G.MG.2
CCR.G.MG.E
STATISTICS AND PROBABILITY (S)
INTERPRETING CATEGORICAL AND QUANTITATIVE DATA (ID)
Summarize, represent, and interpret data on a single count or measurement variable.
5.S.ID.1
Represent data with plots on the real
CC.S.ID.1
Taking political action
number line (dot plots, histograms, and
CCR.S.ID.E
to institute changes in
box plots).
the community
5.S.ID.2
Use statistics appropriate to the shape of
CC.S.ID.2
Understanding there
the data distribution to compare center
are no “true” measures
(median, mean) and spread (interquartile
for statistical quantities
range, standard deviation) of two or more
(i.e., home values,
different data sets.
expected wages,
expected prices,
expected profits or
losses)
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5.S.ID.3
Interpret differences in shape, center, and
spread in the context of the data sets,
accounting for possible effects of extreme
data points (outliers).
5.S.ID.4
CC.S.ID.3
CCR.S.ID.E
Understanding there
are no “true” measures
for statistical quantities
(i.e., home values,
expected wages,
expected prices,
expected profits or
losses)
Tracking one’s daily
expenses personal
financial choices to
data from an ideal
spender’s purchase
log
Use the mean and standard deviation of a CC.S.ID.4
data set to fit it to a normal distribution
ESS04.07.01
and to estimate population percentages.
ESS04.07.02
Recognize that there are data sets for
which such a procedure is not
appropriate. Use calculators,
spreadsheets, and tables to estimate
areas under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables.
5.S.ID.5
Summarize categorical data for two
CC.S.ID.5
Looking at reports on
categories in two-way frequency tables.
CCR.S.ID.E
stock market to see if
Interpret relative frequencies in the
they reflect the trends
context of the data (including joint,
represented
marginal, and conditional relative
frequencies). Recognize possible
associations and trends in the data.*
5.S.ID.6
Represent data on two quantitative
CC.S.ID.6
Tracking monthly
variables on a scatter plot, and describe
ESS04.08.01 savings plan without
how the variables are related.
ESS04.08.02 interest
a. Fit a function to the data; use
functions fitted to data to solve
Plot and analyze
problems in the context of the data.
related student
Use given functions or choose a
attributes (i.e., bed
function suggested by the context.
times, wake up times,
Emphasize linear, quadratic, and
workdays per week,
exponential models.
times eating out per
b. Informally assess the fit of a
week, number of
function by plotting and analyzing
children, hours of
residuals.
sleep per night);
c. Fit a linear function for a scatter
examples of quadratic
plot that suggests a linear
distributions needed
association.
and exponential and
polynomial
distributions
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Interpret linear models.
5.S.ID.7
Interpret the slope (rate of change) and
the intercept (constant term) of a linear
model in the context of the data.
5.S.ID.8
Compute (using technology) and interpret
the correlation coefficient of a linear fit.
5.S.ID.9
Distinguish between correlation and
causation.
CC.S.ID.7
CCR.S.ID.E
CC.S.ID.8
ESS04.08.03
CC.S.ID.9
CCR.S.ID.E
Taking political action
to institute changes in
the community
Taking political action
to institute changes in
the community
Taking political action
to institute changes in
the community
MAKING INFERENCES AND JUSTIFYING CONCLUSIONS (IC)
Understand and evaluate random processes underlying statistical experiments.
5.S.IC.1
Understand statistics as a process for
CC.S.IC.1
Taking political action
making inferences about population
to institute changes in
parameters based on a random sample
the community
from that population.
5.S.IC.2
Decide if a specified model is consistent
CC.S.IC.2
Taking political action
with results from a given data-generating
to institute changes in
process (e.g., using simulation). For
the community
example, a model says a spinning coin
falls heads up with probability 0.5. Would
a result of 5 tails in a row cause you to
question the model?
Make inferences and justify conclusions from sample surveys, experiments, and
observational studies.
5.S.IC.3
Recognize the purposes of and
CC.S.IC.3
differences among sample surveys,
experiments, and observational studies;
explain how randomization relates to
each.
5.S.IC.4
Use data from a sample survey to
CC.S.IC.4
estimate a population mean or proportion;
develop a margin of error through the use
of simulation models for random
sampling.
5.S.IC.5
Use data from a randomized experiment
CC.S.IC.5
to compare two treatments; use
simulations to decide if differences
between parameters are significant.
5.S.IC.6
Evaluate reports based on data.
CC.S.IC.6
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USING PROBABILITY TO MAKE DECISIONS (MD)
Calculate expected values and use them to solve problems.
5.S.MD.1
(+) Define a random variable for a
CC.S.MD.1
quantity of interest by assigning a
numerical value to each event in a sample
space; graph the corresponding
probability distribution using the same
graphical displays as for data
distributions.
5.S.MD.2
(+) Calculate the expected value of a
CC.S.MD.2
random variable; interpret it as the mean
of the probability distribution.
5.S.MD.3
(+) Develop a probability distribution for a CC.S.MD.3
random variable defined for a sample
space in which theoretical probabilities
can be calculated; find the expected
value. For example, find the theoretical
probability distribution for the number of
correct answers obtained by guessing on
all five questions of a multiple-choice test
where each question has four choices,
and find the expected grade under various
grading schemes.
5.S.MD.4
(+) Develop a probability distribution for a CC.S.MD.4
random variable defined for a sample
space in which probabilities are assigned
empirically; find the expected value. For
example, find a current data distribution
on the number of TV sets per household
in the United States, and calculate the
expected number of sets per household.
How many TV sets would you expect to
find in 100 randomly selected
households?
Representing findings
from data gathering in
a manufacturing or
business setting
Gathering data in the
workplace and sorting
it by criteria
Comparing gathered
work-related data by
preparation of
appropriate bar or line
graphs
Comparing gathered
work-related data by
preparation of
appropriate bar or line
graphs
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Use probability to evaluate outcomes of decisions.
5.S.MD.5
(+) Weigh the possible outcomes of a
decision by assigning probabilities to
payoff values and finding expected
values.
a. Find the expected payoff for a
game of chance. For example, find
the expected winnings from a state
lottery ticket or a game at a fast
food restaurant.
b. Evaluate and compare strategies
on the basis of expected values.
For example, compare a highdeductible versus a low-deductible
automobile insurance policy using
various, but reasonable, chances
of having a minor or a major
accident.
5.S.MD.6
(+) Use probabilities to make fair
decisions (e.g., drawing by lots, using a
random number generator).
5.S.MD.7
(+) Analyze decisions and strategies
using probability concepts (e.g., product
testing, medical testing, pulling a hockey
goalie at the end of a game).
CC.S.MD.5
Interpreting the odds
of contracting breast
cancer or being in an
airplane accident
Interpreting the odds
of winning a contest
based on the number
of entries
Interpreting the odds
of being in an accident
CC.S.MD.6
Determining chances
of drawing “short stick”
CC.S.MD.7
Determining the
outcomes of decisions
based on known
variables
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NRS Level 6 Overview
(Please see Overview Explanation of Mathematics NRS Levels 5 & 6 for further
information.)
Number and Quantity (N)
Quantities (Q)
 Reason quantitatively and use units to solve problems
The Complex Number System (CN)
 Perform arithmetic operations with complex numbers
 Represent complex numbers and their operations on the complex plane
 Use complex numbers in polynomial identities and equations
Vector and Matrix Quantities (VM)
 Represent and model with vector quantities
 Perform operations on vectors
 Perform operations on matrices and use matrices in applications
Algebra (A)
Seeing Structure in Expressions (SSE)
 Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Expressions (APR)
 Understand the relationship between zeros and factors of polynomials
 Use polynomial identities to solve problems
 Rewrite rational expressions
Reasoning with Equations and Inequalities (REI)
 Understand solving equations as a process of reasoning and explain the
reasoning
 Solve systems of equations
 Represent and solve equations and inequalities graphically
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NRS Level 6 Overview (continued)
Functions (F)
Interpreting Functions (IF)
 Understand the concept of a function and use function notation
 Interpret functions that arise in applications in terms of the context
 Analyze functions using different representations
Building Functions (BF)
 Build a function that models a relationship between two quantities
 Build new functions from existing functions
Linear, Quadratic, and Exponential Models (LE)
 Construct and compare linear, quadratic, and exponential models and solve
problems
 Interpret expressions for functions in terms of the situation they model
Trigonometric Functions (TF)
 Extend the domain of trigonometric functions using the unit circle
 Model periodic phenomena with trigonometric functions
 Prove and apply trigonometric identities
Geometry (G)
Similarity, Right Triangles, and Trigonometry (SRT)
 Define trigonometric ratios and solve problems involving right triangles
 Apply trigonometry to general triangles
Expressing Geometric Properties with Equations (GPE)
 Translate between the geometric description and the equation for a conic section
 Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension (GMD)
 Explain volume formulas and use them to solve problems
 Visualize relationships between two-dimensional and three-dimensional objects
Modeling with Geometry (MG)
 Apply geometric concepts in modeling situations
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NRS Level 6 Overview (continued)
Statistics and Probability (S)
Conditional Probability and the Rules of Probability (CP)
 Understand independence and conditional probability and use them to interpret
data
 Use the rules of probability to compute probabilities of compound events in a
uniform probability model
Mathematical Practices
1. Make sense of problems
and persevere in solving
them.
★
+
Modeling
Provides foundations for advanced college courses
2. Reason abstractly and
quantitatively.
3. Construct viable arguments
and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
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NRS Level 6 – High Adult Secondary Education
(Grade Levels 11.0-12.9)
Standard
Number
MATH STANDARD
Reference
Example of How
Adults Use This
Skill
NUMBER AND QUANTITY (N)
QUANTITIES★ (Q)
Reason quantitatively and use units to solve problems.
6.N.Q.1
Define appropriate quantities for the purpose of
descriptive modeling.
6.N.Q.2
Choose a level of accuracy appropriate to
limitations on measurement when reporting
quantities.
CC.N.Q.2
CC.N.Q.3
CCR.N.Q.E
THE COMPLEX NUMBER SYSTEM (CN)
Perform arithmetic operations with complex numbers.
6.N.CN.1
CC.N.CN.1
Know there is a complex number i such that 𝑖 2
= –1, and every complex number has the form
a + bi with a and b real.
6.N.CN.2
CC.N.CN.2
Use the relation 𝑖 2 = –1 and the commutative,
associative, and distributive properties to add,
subtract, and multiply complex numbers.
6.N.CN.3
(+) Find the conjugate of a complex number;
CC.N.CN.3
use conjugates to find moduli and quotients of
complex numbers.
Represent complex numbers and their operations on the complex plane.
6.N.CN.4
(+) Represent complex numbers on the
CC.N.CN.4
complex plane in rectangular and polar form
(including real and imaginary numbers), and
explain why the rectangular and polar forms of
a given complex number represent the same
number.
6.N.CN.5
(+) Represent addition, subtraction,
CC.N.CN.5
multiplication, and conjugation of complex
numbers geometrically on the complex plane;
use properties of this representation for
3
computation. For example, (−1 + √3𝑖) = 8
because(−1 + √3𝑖) has modulus 2 and
argument 120°.
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6.N.CN.6
(+) Calculate the distance between numbers in
CC.N.CN.6
the complex plane as the modulus of the
difference, and the midpoint of a segment as
the average of the numbers at its endpoints.
Use complex numbers in polynomial identities and equations.
6.N.CN.7
Solve quadratic equations with real coefficients CC.N.CN.7
that have complex solutions.
6.N.CN.8
(+) Extend polynomial identities to the complex CC.N.CN.8
numbers. For example, rewrite 𝑥 2 + 4 as
(𝑥 + 2𝑖)(𝑥 − 2𝑖).
6.N.CN.9
(+) Know the Fundamental Theorem of Algebra; CC.N.CN.9
show that it is true for quadratic polynomials.
VECTOR AND MATRIX QUANTITIES (VM)
Represent and model with vector quantities.
6.N.VM.1
(+) Recognize vector quantities as having both
magnitude and direction. Represent vector
quantities by directed line segments, and use
appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
6.N.VM.2
(+) Find the components of a vector by
subtracting the coordinates of an initial point
from the coordinates of a terminal point.
6.N.VM.3
(+) Solve problems involving velocity and other
quantities that can be represented by vectors.
Perform operations on vectors.
6.N.VM.4
(+) Add and subtract vectors.
a. Add vectors end-to-end, componentwise, and by the parallelogram rule.
Understand that the magnitude of a sum
of two vectors is typically not the sum of
the magnitudes.
b. Given two vectors in magnitude and
direction form, determine the magnitude
and direction of their sum.
c. Understand vector subtraction v – w as v
+ (–w), where –w is the additive inverse
of w, with the same magnitude as w and
pointing in the opposite direction.
Represent vector subtraction graphically
by connecting the tips in the appropriate
order, and perform vector subtraction
component-wise.
CC.N.VM.1
Studying vector
forces on an object
(e.g., in physics)
CC.N.VM.2
Understanding how
weather affects
flight times and
paths
Studying vector
forces on an object
(e.g., in physics)
CC.N.VM.3
CC.N.VM.4
Studying vector
forces on an object
(e.g., in physics)
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6.N.VM.5
(+) Multiply a vector by a scalar.
CC.N.VM.5
a. Represent scalar multiplication
graphically by scaling vectors and
possibly reversing their direction;
perform scalar multiplication componentwise, e.g., as c(𝑣𝑥 , 𝑣𝑦 ) =(𝑐𝑣𝑥 , 𝑐𝑣𝑦 ) .
b. Compute the magnitude of a scalar
multiple cv using ||cv|| = |c|v. Compute
the direction of cv knowing that when
|c|v ≠ 0, the direction of cv is either along
v (for c > 0) or against v (for c < 0).
Perform operations on matrices and use matrices in applications.
6.N.VM.6
(+) Use matrices to represent and manipulate
CC.N.VM.6
data (e.g., to represent payoffs or incidence
relationships in a network).
6.N.VM.7
(+) Multiply matrices by scalars to produce new CC.N.VM.7
matrices (e.g., as when all of the payoffs in a
game are doubled).
6.N.VM.8
(+) Add, subtract, and multiply matrices of
CC.N.VM.8
appropriate dimensions.
6.N.VM.9
(+) Understand that, unlike multiplication of
CC.N.VM.9
numbers, matrix multiplication for square
matrices is not a commutative operation, but
still satisfies the associative and distributive
properties.
6.N.VM.10 (+) Understand that the zero and identity
CC.N.VM.10
matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the
real numbers. The determinant of a square
matrix is nonzero if and only if the matrix has a
multiplicative inverse.
6.N.VM.11 (+) Multiply a vector (regarded as a matrix with
CC.N.VM.11
one column) by a matrix of suitable dimensions
to produce another vector. Work with matrices
as transformations of vectors.
6.N.VM.12 (+) Work with 2x2 matrices as transformations
CC.N.VM.12
of the plane, and interpret the absolute value of
the determinant in terms of area.
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ALGEBRA (A)
SEEING STRUCTURE IN EXPRESSIONS (SSE)
Write expressions in equivalent forms to solve problems.
6.A.SSE.1 Choose and produce an equivalent form of an
CC.A.SSE.3
expression to reveal and explain properties of
ESS04.08.01
★
the quantity represented by the expression.
a. Complete the square in a quadratic
expression to reveal the maximum or
minimum value of the function it defines.
b. Use the properties of exponents to
transform expressions for exponential
functions. For example the expression
1.15𝑡 can be rewritten as (1.151⁄12 )12𝑡 ≈
1.01212𝑡 to reveal the approximate
equivalent monthly interest rate if the
annual rate is 15%.
6.A.SSE.2 Derive the formula for the sum of a finite
CC.A.SSE.4
geometric series (when the common ratio is not
1), and use the formula to solve problems. For
example, calculate mortgage payments.★
Converting annual
interest to monthly
interest
Understanding
exponential
growth of bacteria
or virus such as
HIV
Converting annual
interest to monthly
interest
ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS (APR)
Understand the relationship between zeros and factors of polynomials.
6.A.APR.1 Know and apply the Remainder Theorem: For a CC.A.APR.2
polynomial p(x) and a number a, the remainder
on division by x – a is p(a), so p(a) = 0 if and
only if (x – a) is a factor of p(x).
6.A.APR.2 Identify zeros of polynomials when suitable
CC.A.APR.3
factorizations are available, and use the zeros
to construct a rough graph of the function
defined by the polynomial.
Use polynomial identities to solve problems.
6.A.APR.3 Prove polynomial identities and use them to
CC.A.APR.4
describe numerical relationships. For example,
the polynomial identity (𝑥 2 + 𝑦 2 )2 = (𝑥 2 −
𝑦 2 )2 + (2𝑥𝑦)2 can be used to generate
Pythagorean triples.
6.A.APR.4 (+) Know and apply the Binomial Theorem for
CC.A.APR.5
𝑛
the expansion of (𝑥 + 𝑦) in powers of x and y
for a positive integer n, where x and y are any
numbers, with coefficients determined for
example by Pascal’s Triangle.
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Rewrite rational expressions.
6.A.APR.5 Rewrite simple rational expressions in different CC.A.APR.6
forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), CCR.A.APR.E
where a(x), b(x), q(x), and r(x) are polynomials
with the degree of r(x) less than the degree of
b(x), using inspection, long division, or, for the
more complicated examples, a computer
algebra system.
6.A.APR.6 (+) Understand that rational expressions form a CC.A.APR.7
system analogous to the rational numbers,
closed under addition, subtraction,
multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide
rational expressions.
REASONING WITH EQUATIONS AND INEQUALITIES (REI)
Understand solving equations as a process of reasoning and explain the reasoning.
6.A.REI.1
Solve simple rational and radical equations in
CC.A.REI.2
one variable, and give examples showing how
NETS•S 4d
extraneous solutions may arise.
Solve systems of equations.
6.A.REI.2
Solve a simple system consisting of a linear
CC.A.REI.7
equation and a quadratic equation in two
variables algebraically and graphically. For
example, find the points of intersection between
the line y = –3x and the circle 𝑥 2 + 𝑦 2 = 3.
6.A.REI.3
(+) Represent a system of linear equations as a CC.A.REI.8
single matrix equation in a vector variable.
6.A.REI.4
(+) Find the inverse of a matrix if it exists, and
CC.A.REI.9
use it to solve systems of linear equations
(using technology for matrices of dimension 3 ×
3 or greater).
Represent and solve equations and inequalities graphically.
6.A.REI.5
Explain why the x-coordinates of the points
CC.A.REI.11
where the graphs of the equations y = f(x) and y ESS04.08.02
= g(x) intersect are the solutions of the equation ESS04.08.03
f(x) = g(x); find the solutions approximately
(e.g., using technology to graph the functions,
make tables of values, or find successive
approximations). Include cases where f(x)
and/or g(x) are linear, polynomial, rational,
absolute value, exponential, and logarithmic
functions.★
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6.A.REI.6
Graph the solutions to a linear inequality in two
variables as a half-plane (excluding the
boundary in the case of a strict inequality), and
graph the solution set to a system of linear
inequalities in two variables as the intersection
of the corresponding half-planes.
CC.A.REI.12
FUNCTIONS (F)
INTERPRETING FUNCTIONS (IF)
Understand the concept of a function and use function notation.
6.F.IF.1
Understand that a function from one set (called
CC.F.IF.1
the domain) to another set (called the range)
CCR.F.IF.E
assigns to each element of the domain exactly
one element of the range. If f is a function and x
is an element of its domain, then f(x) denotes the
output of f corresponding to the input x. The
graph of f is the graph of the equation y = f(x).
6.F.IF.2
Use function notation, evaluate functions for
CC.F.IF.2
inputs in their domains, and interpret statements ESS04.08.03
that use function notation in terms of a context.
CCR.F.IF.E
6.F.IF.3
Recognize that sequences are functions,
CC.F.IF.3
sometimes defined recursively, whose domain is ESS04.08.03
a subset of the integers. For example, the
Fibonacci sequence is defined recursively by f(0)
= f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context.
6.F.IF.4
For a function that models a relationship
CC.F.IF.4
between two quantities, interpret key features of
ESS01.03.06
graphs and tables in terms of the quantities, and CCR.F.IF.E
sketch graphs showing key features given a
verbal description of the relationship. Key
features include: intercepts; intervals where the
function is increasing, decreasing, positive, or
negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★
6.F.IF.5
Relate the domain of a function to its graph and, CC.F.IF.5
where applicable, to the quantitative relationship CCR.F.IF.E
it describes. For example, if the function h(n)
gives the number of person-hours it takes to
assemble n engines in a factory, then the
positive integers would be an appropriate domain
for the function.★
Describing
change in
workplace output
relative to
contributing
factors (i.e.,
number of
employees, shift
time, time of
year)
Reading a chart
or graph in a
health pamphlet
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6.F.IF.6
Calculate and interpret the average rate of
change of a function (presented symbolically or
as a table) over a specified interval. Estimate the
rate of change from a graph.★
Analyze functions using different representations.
6.F.IF.7
Graph functions expressed symbolically and
show key features of the graph, by hand in
simple cases and using technology for more
complicated cases. ★
a. Graph square root, cube root, and
piecewise-defined functions, including
step functions and absolute value
functions.
b. Graph polynomial functions, identifying
zeros when suitable factorizations are
available, and showing end behavior.
c. (+) Graph rational functions, identifying
zeros and asymptotes when suitable
factorizations are available, and showing
end behavior.
d. Graph exponential and logarithmic
functions, showing intercepts and end
behavior, and trigonometric functions,
showing period, midline, and amplitude.
6.F.IF.8
Write a function defined by an expression in
different but equivalent forms to reveal and
explain different properties of the function.
a. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
and symmetry of the graph, and interpret
these in terms of a context.
b. Use the properties of exponents to
interpret expressions for exponential
functions. For example, identify percent
rate of change in functions such as 𝑦 =
CC.F.IF.6
CCR.F.IF.E
Conversing about
information in
newspapers and
magazines
CC.F.IF.7
ESS01.03.06
Scatter plot and
analyze related
student attributes
(i.e., examples of
exponential and
polynomial
distributions
needed)
CC.F.IF.8
CCR.F.IF.E
Tracking monthly
savings plan
without interest
(1.02)𝑡 , 𝑦 = (0.97)𝑡 , 𝑦 = (1.01)12𝑡 , 𝑦 =
𝑡
(1.2) ⁄10 , and classify them as representing
exponential growth or decay.
6.F.IF.9
Compare properties of two functions each
represented in a different way (algebraically,
graphically, numerically in tables, or by verbal
descriptions). For example, given a graph of one
quadratic function and an algebraic expression
for another, say which has the larger maximum.
CC.F.IF.9
CCR.F.IF.E
Plot and analyze
related student
attributes (i.e.,
bed times and
wake up times,
workdays per
week and times
eating out per
week, number of
children, number
of hours of sleep
per night)
Reading a graph
in an ad or poster
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BUILDING FUNCTIONS (BF)
Build a function that models a relationship between two quantities.
6.F.BF.1 Combine standard function types using arithmetic CC.F.BF.1
operations.
a. (+) Compose functions. For example, if
T(y) is the temperature in the
atmosphere as a function of height,
and h(t) is the height of a weather
balloon as a function of time, then
T(h(t)) is the temperature at the
location of the weather balloon as a
function of time.
6.F.BF.2
Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them
to model situations, and translate between the
two forms.★
Build new functions from existing functions.
6.F.BF.3 Identify the effect on the graph of replacing f(x)
by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the
value of k given the graphs. Experiment with
cases and illustrate an explanation of the effects
on the graph using technology. Include
recognizing even and odd functions from their
graphs and algebraic expressions for them.
6.F.BF.4 Find inverse functions.
a. Solve an equation of the form f(x) = c for a
simple function f that has an inverse and
write an expression for the inverse. For
example, 𝑓(𝑥) = 2𝑥 3 𝑜𝑟 𝑓(𝑥) = (𝑥 + 1)/(𝑥 −
CC.F.BF.2
CC.F.BF.3
NETS•S 4d
CC.F.BF.4
6.F.BF.5
Entering an
expression in a
spreadsheet
Keeping personal
finance records
1) 𝑓𝑜𝑟 𝑥 ≠ 1
b. (+) Verify by composition that one function
is the inverse of another.
c. (+) Read values of an inverse function
from a graph or a table, given that the
function has an inverse.
d. (+) Produce an invertible function from a
non-invertible function by restricting the
domain.
(+) Understand the inverse relationship between
exponents and logarithms and use this
relationship to solve problems involving
logarithms and exponents.
Building a
function that
models the
temperature of a
cooling body by
adding a constant
function to a
decaying
exponential, and
relating these
functions to the
model
Keeping personal
finance records
CC.F.BF.5
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LINEAR, QUADRATIC, AND EXPONENTIAL MODELS★ (LE)
Construct and compare linear, quadratic, and exponential models and solve problems.
6.F.LE.1 Distinguish between situations that can be
CC.F.LE.1
Understanding
modeled with linear functions and with
CCR.F.LE.E
the importance of
exponential functions.
investing early to
a. Prove that linear functions grow by equal
take advantage of
differences over equal intervals, and that
compounded
exponential functions grow by equal
returns (i.e.,
factors over equal intervals.
college or
b. Recognize situations in which a quantity
retirement)
grows or decays by a constant percent
rate per unit interval relative to another.
6.F.LE.2 Observe using graphs and tables that a quantity
CC.F.LE.3
Discussing with a
increasing exponentially eventually exceeds a
financial planner
quantity increasing linearly, quadratically, or
about retirement
(more generally) as a polynomial function.
investment plans
offered at work
6.F.LE.3 For exponential models, express as a logarithm
CC.F.LE.4
the solution to 𝑎𝑏 𝑐𝑡 = 𝑑 where a, c, and d are
numbers and the base b is 2, 10, or e; evaluate
the logarithm using technology.
Interpret expressions for functions in terms of the situation they model.
6.F.LE.4 Interpret the parameters in a linear or exponential CC.F.LE.5
Setting realistic
function in terms of a context.
CCR.F.LE.E
investment goals
(i.e., college or
retirement)
TRIGONOMETRIC FUNCTIONS (TF)
Extend the domain of trigonometric functions using the unit circle.
6.F.TF.1
Understand radian measure of an angle as the CC.F.TF.1
length of the arc on the unit circle subtended
by the angle.
6.F.TF.2
Explain how the unit circle in the coordinate
CC.F.TF.2
plane enables the extension of trigonometric
functions to all real numbers, interpreted as
radian measures of angles traversed
counterclockwise around the unit circle.
6.F.TF.3
(+) Use special triangles to determine
CC.F.TF.3
geometrically the values of sine, cosine,
tangent for 𝜋⁄3 , 𝜋⁄4 , 𝑎𝑛𝑑 𝜋/6, and use the unit
circle to express the values of sine, cosine,
and tangent for 𝜋 − 𝑥, 𝜋 + 𝑥, 𝑎𝑛𝑑 2𝜋 − 𝑥 in
terms of their values for x, where x is any real
number.
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6.F.TF.4
(+) Use the unit circle to explain symmetry
(odd and even) and periodicity of trigonometric
functions.
Model periodic phenomena with trigonometric functions.
6.F.TF.5
Choose trigonometric functions to model
periodic phenomena with specified amplitude,
frequency, and midline.★
CC.F.TF.4
6.F.TF.6
CC.F.TF.6
(+) Understand that restricting a trigonometric
function to a domain on which it is always
increasing or always decreasing allows its
inverse to be constructed.
6.F.TF.7
(+) Use inverse functions to solve trigonometric
equations that arise in modeling contexts;
evaluate the solutions using technology, and
interpret them in terms of the context.★
Prove and apply trigonometric identities.
6.F.TF.8
Prove the Pythagorean identity 𝑠𝑖𝑛2 (𝜃) +
𝑐𝑜𝑠 2 (𝜃) = 1 and use it to find
sin(𝜃), cos(𝜃) , 𝑜𝑟 tan(𝜃) given sin(θ), cos(θ), or
tan(θ) and the quadrant of the angle.
6.F.TF.9
(+) Prove the addition and subtraction formulas
for sine, cosine, and tangent and use them to
solve problems.
CC.F.TF.5
Creating a
trigonometric
function to model
heating costs
throughout the
year
CC.F.TF.7
CC.F.TF.8
CC.F.TF.9
GEOMETRY (G)
SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY (SRT)
Define trigonometric ratios and solve problems involving right triangles.
6. G.SRT.1 Understand that by similarity, side ratios in
CC.G.SRT.6
right triangles are properties of the angles in
the triangle, leading to definitions of
trigonometric ratios for acute angles.
6.G.SRT.2
Explain and use the relationship between the
CC.G.SRT.7
sine and cosine of complementary angles.
6.G.SRT.3
Use trigonometric ratios and the Pythagorean
CC.G.SRT.8
Theorem to solve right triangles in applied
problems.★
Apply trigonometry to general triangles.
(+) Derive the formula A = 1/2 ab sin(C) for the CC.G.SRT.9
6.G.SRT.4
area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite
side.
Building and
designing
structures
Designing
products
Designing
products
Designing
products
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6.G.SRT.5
(+) Prove the Laws of Sines and Cosines and
use them to solve problems.
CC.G.SRT.10
Designing
products
6.G.SRT.6
(+) Understand and apply the Law of Sines
and the Law of Cosines to find unknown
measurements in right and non-right triangles
(e.g., surveying problems, resultant forces).
CC.G.SRT.11
Designing
products
EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS (GPE)
Translate between the geometric description and the equation for a conic section.
6.G.GPE.1 Derive the equation of a circle of given center
CC.G.GPE.1
Using
and radius using the Pythagorean Theorem;
CAD/CAM
complete the square to find the center and
software to
radius of a circle given by an equation.
design a
product
6.G.GPE.2 Derive the equation of a parabola given a
CC.G.GPE.2
Reading scientific
focus and directrix.
diagrams
6.G.GPE.3 (+) Derive the equations of ellipses and
CC.G.GPE.3
Using CAD/CAM
hyperbolas given the foci, using the fact that
software to design
the sum or difference of distances from the foci
a product
is constant.
Use coordinates to prove simple geometric theorems algebraically.
6.G.GPE.4 Use coordinates to prove simple geometric
CC.G.GPE.4
Proving or
theorems algebraically.
disproving that a
figure defined by
four given points in
the coordinate
plane is a
rectangle; proving
or disproving that
the point (1, √3) lies
on the circle
centered at the
origin and
containing the point
(0, 2)
GEOMETRIC MEASUREMENT AND DIMENSION (GMD)
Explain volume formulas and use them to solve problems.
6.G.GMD.1 Give an informal argument for the formulas for CC.G.GMD.1
the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s principle, and
informal limit arguments.
Building and
measuring
structures and
objects
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6.G.GMD.2
(+) Give an informal argument using Cavalieri’s CC.G.GMD.2
principle for the formulas for the volume of a
sphere and other solid figures.
Building and
measuring
structures and
objects
Visualize relationships between two-dimensional and three-dimensional objects.
6.G.GMD.3 Identify the shapes of two-dimensional crossCC.G.GMD.4 Building and
sections of three dimensional objects, and
measuring
identify three-dimensional objects generated
structures and
by rotations of two-dimensional objects.
objects
MODELING WITH GEOMETRY (MG)
Apply geometric concepts in modeling situations.
6.G.MG.1
Apply geometric methods to solve design
problems (e.g., designing an object or
structure to satisfy physical constraints or
minimize cost; working with typographic grid
systems based on ratios).★
CC.G.MG.3
Building and
measuring
structures and
objects
CONDITIONAL PROBABILITY AND THE RULES OF PROBABILITY (CP)
Understand independence and conditional probability and use them to interpret data.
6.S.CP.1
Describe events as subsets of a sample space CC.S.CP.1
Tossing a coin or
(the set of outcomes) using characteristics (or
rolling a die
categories) of the outcomes, or as unions,
intersections, or complements of other events
(“or,” “and,” “not”).
6.S.CP.2
Understand that two events A and B are
CC.S.CP.2
Tossing a coin or
independent if the probability of A and B
rolling a die
occurring together is the product of their
probabilities, and use this characterization to
determine if they are independent.
6.S.CP.3
Understand the conditional probability of A
CC.S.CP.3
Tossing a coin or
given B as P(A and B)/P(B), and interpret
rolling a die
independence of A and B as saying that the
conditional probability of A given B is the same
as the probability of A, and the conditional
probability of B given A is the same as the
probability of B.
6.S.CP.4
Construct and interpret two-way frequency
CC.S.CP.4
Estimating the
tables of data when two categories are
ESS01.03.06 probability that a
associated with each object being classified.
randomly selected
Use the two-way table as a sample space to
student from class
decide if events are independent and to
will favor science.
approximate conditional probabilities. For
Do the same for
example, collect data from a random sample of
other subjects and
students in your class on their favorite subject
compare the
among math, science, and English.
results
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6.S.CP.5
Recognize and explain the concepts of
conditional probability and independence in
everyday language and everyday situations.
CC.S.CP.5
Comparing the
chance of having
lung cancer if you
are a smoker with
the chance of
being a smoker if
you have lung
cancer
Use the rules of probability to compute probabilities of compound events in a uniform
probability model.
6.S.CP.6
Find the conditional probability of A given B as CC.S.CP.6
Tossing a coin or
the fraction of B’s outcomes that also belong to
rolling a die
A, and interpret the answer in terms of the
model.
6.S.CP.7
Apply the Addition Rule, P(A or B) = P(A) +
CC.S.CP.7
Tossing a coin or
P(B) – P(A and B), and interpret the answer in
rolling a die
terms of the model.
6.S.CP.8
(+) Apply the general Multiplication Rule in a
CC.S.CP.8
Tossing a coin or
uniform probability model, P(A and B) =
rolling a die
P(A)P(B|A) = P(B)P(A|B), and interpret the
answer in terms of the model.
6.S.CP.9
(+) Use permutations and combinations to
CC.S.CP.9
Tossing a coin or
compute probabilities of compound events and
rolling a die
solve problems.
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Illinois ABE/ASE Content Standards
Math Glossary
A
AA similarity: Angle-angle similarity. When two triangles have corresponding angles
that are congruent, the triangles are similar.
Absolute value: a) A nonnegative number equal in numerical value to a given real
number; b) The distance from a number to zero on the number line.
Acute angle: Measures less than 90°.
Acute triangle: Has three acute angles.
Addition and subtraction within 5, 10, 20, 100, or 1000: Addition or subtraction of two
whole numbers with whole number answers, and with sum or minuend in the range 0-5,
0-10, 0-20, or 0-100, respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 =
9 is a subtraction within 20, and 55 – 18 = 37 is a subtraction within 100.
Additive inverses: Two numbers whose sum is 0 are additive inverses of one another.
Example: 3/4 and -3/4 are additive inverses of one another because 3/4 + (-3/4) = (3/4) + (3/4) = 0.
Adjacent angles: Angles that have a common vertex and a common ray.
Algebraic expressions: Like numerical expressions, except that both letters and
numbers can be used in such expressions.
Algorithm: A finite set of steps for completing a procedure (e.g., long division).
Alternate exterior angles: A pair of congruent angles formed by two parallel lines cut
by a transversal, located outside the parallel lines and on opposite sides of the
transversal.
Alternate interior angles: A pair of congruent angles formed by two parallel lines cut
by a transversal, located inside the parallel lines and on opposite sides of the
transversal.
Analog: Data represented by continuous variables (e.g., a clock with hour, minute, and
second hands).
Analytic geometry: The branch of mathematics that uses functions and relations to
study geometric phenomena (e.g., the description of eclipses and other conic sections
in the coordinate plane by quadratic equations).
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Angle: The space or opening between a pair of lines, called rays. A figure formed when
two rays meet at a single point, called the 'vertex'.
Area: Measure of the amount of surface within the perimeter of a flat (two-dimensional)
figure. Area is always measured in 'square' units.
Arithmetic Sequence: A sequence of numbers in which each term except the first
term is the result of adding the same number, called the common difference, to the
preceding term.
ASA congruence: Angle-side-angle congruence; when two triangles have
corresponding angles and sides that are congruent, the triangles themselves are
congruent.
Associative [grouping] property: A mathematical rule stating that when more than
two numbers are added or multiplied, the result will be the same no matter how the
numbers are grouped:
(a + b) = c = a + (b + c) or (a × b) × c = a × (b × c). (See Table 2)
Assumption: A fact of statement (as a proposition, axiom, postulate, or notion) taken
for granted.
Asymptotes: A line that continually approaches a given curve but does not meet it at
any finite distance.
Attribute: A common feature of a set of figures.
Average: Add the items to average, and then divide this total by the number of items in
the list.
B
Bar graph: Visual presentation of data from different sources (e.g., the height or length
of bars against the same scale).
Benchmark fraction: A common fraction against which other fractions can be
measured, such as ½.
Binomial Theorem: A method for distributing powers of binomials.
Bisect: To divide into two equal parts.
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Bivariate data: Pairs of linked numerical observations (e.g., a list of heights and
weights for each player on a football team).
Box plot: A graphic method that shows the distribution of data values by using the
median, quartiles, and extremes of the data set. A box shows the middle 50% of the
data.
C
c = nr = c (total cost) = n (number of items) r (rate or price per item) to find the total
cost of a number of similar items. A formula is a standardized (consistent) equation to
use in order to answer the same type of question.
Calculus: The mathematics of change and motion. The main concepts of calculus are
limits, instantaneous rates of change, and areas enclosed by curves.
Capacity: The maximum amount of number that can be contained or accommodated,
e.g., a just with a one-gallon capacity; the auditorium was filled to capacity.
Cardinal number: A number (as 1, 5, 15) that is used in simple counting and that
indicates how many elements there are in a set.
Cartesian plane: A coordinate plane with perpendicular coordinate axes.
Cavalieri’s principle: A method, with formula given below, of finding the volume of any
solid for which cross-sections by parallel planes have equal areas. This includes, but is
not limited to, cylinders and prisms. Formula: Volume = Bh, where B is the area of a
cross-section and h is the height of the solid.
Chart: The visual organization and presentation of data in rows and columns.
Circle: The curve formed by all the points in a plane that are the same distance (radius)
from a given point (center) and there is 360° in a circle.
Circle graph: A visual presentation of data showing parts of a whole (the circle) using
percents, decimals, or fractions.
Circumference: The distance around the edge of a circle.
Coefficient: Any of the factors of a product considered in relation to a specific factor.
Commutative [order] property: A mathematical rule stating that the order in which
numbers are added (or multiplied) does not change the sum (or product): a + b = b + a
or a × b = b × a. (See Table 3)
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Combination: An arrangement of objects in which order does not matter.
Comparing: Determining which number is greater; arranging numbers in order; using
equality and inequality symbols (=, <, >, <, >).
Complementary angles: Two angles for which the sum of their measures is 90°.
Complex fraction: A fraction A/B where A and/or B are fractions (B nonzero).
Complex number: A number that can be written as the sum or difference of a real
number and an imaginary number. (See Illustration 1)
Complex plane: The coordinate plane used to graph complex numbers.
Compose numbers: a) Given pairs, triples, etc. of numbers, identify sums or products
that can be computed; b) Each place in the base ten place value is composed of ten
units of the place to the left (e.g., one hundred is composed of ten bundles of ten, one
ten is composed of ten ones, etc).
Compose shapes: Join geometric shapes without overlaps to form new shapes.
Composite number: A whole number that has more than two factors.
Computation algorithm: A set of predefined steps applicable to a class of problems
that gives the correct result in every case when the steps are carried out correctly. See
algorithm and computation strategy.
Computation strategy: Purposeful manipulations that may be chosen for specific
problems that gives the correct result in every case when the steps are carried out
correctly. See computation algorithm.
Congruent: Two plane or solid figures are congruent if one can be obtained from the
other by rigid motion (a sequence of rotations, reflections, and translations); having the
same size and shape; equal.
Congruent angles: Angles that have equal measures.
Congruent figures: Figures that have the same shape and size.
Conic Section:
Ellipse: A curved line forming a closed loop, where the sum of the distances
from two points (foci) to every point on the line is constant.
Hyperbola: A symmetrical open curve formed by the intersection of a cone with
a plane at a smaller angle with its axis than the side of the cone.
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Parabola: The set of points that is the same distance away from a single point
called the focus and a line called the directrix. Parabolas are symmetric, and their
lines of symmetry pass through the vertex. The vertex is the point which has the
smallest distance between both the focus and the directrix.
Conjugate: The result of writing sum of two terms as a difference, or vice versa. For
example, the conjugate of x - 2 is x + 2.
Convert: To change.
Conversions: The equivalency of change from one unit of measurement to another.
Coordinate graph: A system for finding the location of a point on a flat surface called a
plane, and two numbers, an x-coordinate and a y-coordinate, name each point on the
grid.
Coordinate plane: A plane in which two coordinate axes are specified (i.e., two
intersecting directed straight lines, usually perpendicular to each other, and usually
called the x-axis and y-axis). Every point in a coordinate plane can be described
uniquely by an ordered pair of numbers, the coordinates of the point with respect to the
coordinate axes.
Coordinate system: A set of points formed by a grid with a horizontal x-axis and
vertical y-axis. The line that extends horizontally in both directions through the origin is
the x-axis. The y-axis extends vertically in both directions through this central point.
Cosine: A trigonometric function that for an acute angle is the ratio between a leg
adjacent to the angle when the angle is considered part of a right triangle and the
hypotenuse.
Counting number: A number used in counting objects (e.g., a number from the set 1,
2, 3, 4, 5….) (See Illustration 1)
Counting on: A strategy for finding the number of objects in a group without having to
count every member of the group. For example, if a stack of books is known to have 8
books and 3 more books are added to the top, it is not necessary to count the stack all
over again. One can find the total by counting on—pointing to the top book and saying
“eight,” following this with “nine, ten, eleven. There are eleven books now.”
Cube: A rectangular solid with six square sides (faces-all edges of equal length); raise
to the third power (3) - multiply a number times itself 3 times( n x n x n = n3).
Cylinder: A solid (three-dimensional) figure with two congruent circular bases and
straight sides.
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D
d is the distance, r is the rate, and t is the time (d = rt).
Decimal: A fraction expressed in the place value system to the right of the decimal
point.
Decimal expansion: Writing a rational number as a decimal.
Decimal fraction: A fraction (as 0.25 = 25/100 or 0.025 = 25/1000) or mixed number
(as 3.025 = 3 25/1000) in which the denominator is a power of ten, usually expressed by
the use of the decimal point.
Decimal number: Any real number expressed in base 10 notation, such as 2.673
Decompose numbers: Given a number, identify pairs, triples, etc. of numbers that
combine to form the given number using subtraction and division.
Decompose shapes: Given the geometric shape, identify geometric shapes that meet
without overlap to form the given shape.
Denominator: The bottom number of a fraction.
Diagonal: A line segment drawn between the vertices of two nonadjacent sides of a
figure that has four or more straight sides.
Diameter: The distance across a circle that passes through the middle.
Dilation: A transformation that moves each point along the ray through the point
emanating from a fixed center, and multiplies distances from the center by a common
scale factor.
Digit: a) Any of the Arabic numerals 1 to 9 and usually the symbol 0; b) One of the
elements that combine to form numbers in a system other than the decimal system.
Directrix: A fixed curve with which a generatrix maintains a given relationship in
generating a geometric figure; specifically: a straight line the distance to which from any
point in a conic section is in fixed ratio to the distance from the same point to a focus.
Distance formula: Distance (d) = rate ( r) x time (t) or d = rt
Distributive property: Means to multiply a factor by a sum of terms, multiply the factor
by each term in parentheses, than combine the products.
Domain: The set of all possible input values (usually x) which allows the function
formula to work.
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Dot plot: See line plot.
E
Equation: The mathematical statement that says two expressions are equal.
Equidistant: A point the same distance from two or more other points.
Equilateral: Having equal sides.
Equilateral triangle: Triangle with three congruent sides; an equilateral triangle also
has three congruent angles, each measuring 60°.
Evaluate: An expression substitute known or given values for the variables in an
algebraic expression (perform the operations in the order of operations to obtain the
solution).
Expanded form: A multi-digit number is expressed in expanded form when it is written
as a sum of single-digit multiples of powers of ten. For example: 643 = 600 + 40 + 3
Expected value: For a random variable, the weighted average of its possible values,
with weights given by their respective probabilities.
Exponent: The number that indicates how many times the base is used as a factor
(e.g., in 43 = 4 x 4 x 4 = 64, the exponent is 3, indicating that 4 is repeated as a factor
three times).
Exponential function: A function of the form y = a ∙ bx where a > 0 and either 0 < b < 1
or b > 1. The variables do not have to be x and y. For example, A = 3.2 ∙ (1.02)t is an
exponential function.
Expression: A mathematical phrase that combines operations, numbers, and/or
variables (e.g., 32 - a).
F
Factor: To find the number of elements that exist within that particular number.
Factoring: Finding the algebraic terms or expressions (called factors) that when
multiplied will result in a certain product.
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First quartile: For a data set with median M, the first quartile is the median of the data
values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the
first quartile is 6.2 See median, third quartile, and interquartile range.
FOIL: A method for multiplying algebraic factors with more than one term. FOIL stands
for first-outer-inner-last.
Formula: An equation that helps solve math problems.
Fraction: A way to show part (the numerator or top number) of a whole (the
denominator or bottom number). Digits grouped above and below a division bar is a
ratio.
Function: An algebraic rule involving two variables in which for every value of the first
variable (x), there is a unique value of the second variable (y).
Function notation: A notation that describes a function. For a function f, when x is a
member of the domain, the symbol f(x) denotes the corresponding member of the range
(e.g., f(x) = x + 3).
Fundamental Theorem of Algebra: The theorem that establishes that, using complex
numbers, all polynomials can be factored. A generalization of the theorem asserts that
any polynomial of degree n has exactly n zeros, counting multiplicity.
G
Geometric sequence (progression): An ordered list of numbers that has a common
ratio between consecutive terms (e.g., 2, 6, 18, 54.).
Graph: A visual representation comparing data from different sources or over time.
H
Histogram: A type of bar graph used to display the distribution of measurement data
across a continuous range.
Hypotenuse: The side of a right triangle opposite the right angle. The hypotenuse is
always the longest side.
Horizontal axis: A scale that runs along the bottom or left to right on a graph or
coordinate grid; the x-axis.
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Hypotenuse of a right triangle: The line that is opposite the right angle. It is
represented by the letter c in the formula c2 = a2 + b2.
I
Identity property of 0: The sum of zero and any number is that given number. (See
Table 3)
Imaginary number: Complex numbers with no real terms, such as 5i. (See Illustration
1)
Improper fraction: A fraction in which the value of the numerator is greater than or
equal to the value of the denominator.
Independently combined probability models: Two probability models are said to be
combined independently if the probability of each ordered pair in the combined model
equals the product of the original probabilities of the two individual outcomes in the
ordered pair.
Inequalities: Statements that two amounts are not equal.
Integers: All whole numbers, positive, negative, and zero.
Intercept: The point where the line crosses the y-axis. This is found by setting x = O.
Interest: The amount of money that is paid or charged for the use of money over a
certain period of time. The simple interest formula is (p)rincipal x (r)ate x (t)ime or
interest = prt.
Interquartile range: A measure of variation in a set of numerical data, the interquartile
range is the distance between the first and third quartiles of the data set. Example: For
the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15 – 6 = 9.
See first quartile and third quartile.
Inverse: Opposite; addition and subtraction are inverse operations, as are multiplication
and division.
Inverse function: A function obtained by expressing the dependent variable of one
function as the independent variable of another; that is the inverse of y – f(x) is x = f –
1(y).
Irrational number: A number that cannot be expressed as a quotient of two integers
(e.g., √2). It can be shown that a number is irrational if and only if it cannot be written
as a repeating or terminating decimal.
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Isosceles: Having two sides equal.
Isosceles triangle: A triangle in which two sides have the same length, and the two
angles opposite the equal sides have the same measure.
L
Law of Cosines: An equation relating the cosine of an interior angle and the lengths of
the sides of a triangle.
Law of Sines: Equations relating the sines of the interior angles of a triangle and the
corresponding opposite sides.
Leg in a right triangle: One of the two sides that form the right angle.
Like terms: The same variable or variables raised to the same power.
Line graph: Visual presentation of data as a line on a grid, often showing change over
time (a trend).
Linear association: Two variables have a linear association if a scatter plot of the data
can be well-approximated by a line.
Linear equation: An equation that does not contain a variable to any power (exponent)
greater than 1; an equation whose graph is a straight line.
Line plot: A method of visually displaying a distribution of data values where each data
value is shown as a dot or mark above a number line; also known as a dot plot.
Linear function: A mathematical function in which the variables appear only in the first
degree, are multiplied by constants, and are combined only by addition and subtraction,
multiplied by constants, and are combined only by addition and subtraction. For
example: f(s) = Ax + By +C
Logarithm: The exponent that indicates the power to which a base number is raised to
produce a given number. For example, the logarithm of 100 to the base 10 is 2.
Logarithmic function: Any function in which an independent variable appears in the
form of a logarithm; they are the inverse functions of exponential functions.
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M
Magnitude: The length of the vector.
Matrix: A rectangular array of numbers or variables. Plural: matrices.
Maxima: y0 is the "absolute maximum" of f(x) on I if and only if y0 >= f(x) for all x on I.
Mean: A measure of center in a set of numerical data, computed by adding the values
in a list and then dividing by the number of values in the list, the average. Example: For
the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.
Mean absolute deviation: A measure of variation in a set of numerical data, computed
by adding the distances between each data value and the mean, then dividing by the
number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120},
the mean absolute deviation is 20.
Measure of variability: A determination of how much the performance of a group
deviates from the mean or median; most frequently used measure is standard deviation.
Median: A measure of center in a set of numerical data. The median of a list of values
is the value appearing at the center of a sorted version of the list; or the mean of the two
central values, if the list contains an even number of values. Example: For the data set
{2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.
Metric system of measurement: A measurement system used throughout most of the
world that is based on the powers of ten. Common units are meters (unit of length),
grams (unit of weight or mass), and liters (unit of volume).
Midline: In the graph of a trigonometric function, the horizontal line halfway between its
maximum and minimum values. Multiplication and division within 100. Multiplication or
division of two whole numbers with whole number answers, and with product or
dividend in the range 0-100. Example: 72 ÷ 8 = 9.
Minima: y0 is the "absolute minimum" of f(x) on I if and only if y0 <= f(x) for all x on I.
Mixed number: A quantity expressed as a whole number and a proper fraction.
Mode: A list of data is the number occurring most often.
Model: A mathematical representation (e.g., number, graph, matrix, equation(s),
geometric figure) for real-world or mathematical objects, properties, actions, or
relationships.
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Modulus of a complex number: The distance between a complex number and the
origin on the complex plane. The absolute value of a + bi is written |a + bi|, and the
formula for |a + bi| is a 2  b2 . For a complex number in polar form, r(cos + i sin ), the
modulus is r.
Multiplicationand division within 100: The multiplication or division of two whole
numbers with whole number answers, and with product or dividend in the range
0-100. Example: 72 ÷ 8 = 9.
Multiplicative inverses: Two numbers whose product is 1 are multiplicative inverses
of one another. Example: 3/4 and 4/3 are multiplicative inverses of one another
because 3/4 x 4/3 = 4/3 x 3/4 = 1.
N
Negative number: A number to the left of zero on a number line; a number less than
zero in value; used to show a decrease, a loss, or downward direction; may be
preceded by a minus sign.
Network: a) A figure consisting of vertices and edges that shows how objects are
connected, b) A collection of points (vertices), with certain connections (edges) between
them.
Non-adjacent angles: Angles that do not share a common ray; they may or may not
share a common vertex.
Number line: A line divided into equal segments (intervals) by points corresponding to
integers, fractions, or decimals; points to the right of 0 are positive; those to the left are
negative. Signed numbers extend indefinitely in both directions on the number line.
Number line diagram: A diagram of the number line used to represent numbers and
support reasoning about them. In a number line diagram for measurement quantities,
the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.
Numeral: A symbol or mark used to represent a number.
Numerator: The top number in a fraction.
O
Obtuse angle: Measures more than 90° but less than 180°.
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Order of Operation: The part of a mathematical expression that must be computed
first, second, third, and so on. It is the sequence, agreed upon by mathematicians, for
performing mathematical operations: 1. Operations in grouping symbols, 2. Exponents
and roots, 3. Multiplication and division from left to right, 4. Addition and subtraction
from left to right.
Order pair: A pair is of numbers that names a point on a coordinate graph; presented in
parentheses (the x- coordinate, the y-coordinate).
Ordinal number: A number designating the place (as first, second, or third, etc.)
occupied by an item in an ordered sequence.
Operation: An action preformed to one or more numbers to produce an answer:
addition, subtraction, multiplication, division, exponents, and roots.
Origin: The point at which the x-axis and y-axis in a coordinate graph intersect; the
point represented by the ordered pair (0,0).
Outlier: An element of a data set that distinctly stands out from the rest of the data; it is
significantly larger or smaller than the other numbers in the data set.
P
Parallel lines: Two lines on the same plane that do not intersect.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel; opposite sides
are of equal length, and opposite angles are of equal measure.
Parentheses: What surrounds the mathematical expression (24 - 16) + 67 = 75.
Partition: A process of dividing an object into parts.
Pascal’s triangle: A triangular arrangement of numbers in which each row
starts and ends with 1, and each other number is the sum of the two numbers
above it.
Percent: The number of parts per hundred. A percent is another way of expressing a
fraction with a denominator of 100.
Percent rate of change: A rate of change expressed as a percent. Example: If a
population grows from 50 to 55 in a year, it grows by 5/50 = 10% per year.
Period (of trig functions): The distance between any two repeating points on the
function.
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Periodic phenomena: Naturally recurring events. For example, ocean tides and
machine cycles.
Perimeter: The distance around a flat (2-D) figure; the sum of the lengths of all the
sides of a flat figure.
Permutation: Different arrangements of a group of items where order matters.
Perpendicular lines: Two lines that intersect forming adjacent right angles.
Pi (𝝅): The constant ratio of the circumference of a circle to the diameter; approximately
3.14 or 22/7.
Picture graph: A graph that uses pictures to show and compare information.
Plane: A set of points that forms a flat surface.
Polynomial: The sum or difference of terms which have variables raised to positive
integer powers and which have coefficients that may be real or complex. The following
are all polynomials: 5x3 - 2x2 + x - 13, x2y3 + xy, and (1 + I)a2 + ib2
Polynomial function: Any function whose value is the solution of a polynomial.
Positive number: A number to the right of zero on a number line; a number greater
than zero in value, used to show an increase, a gain, or upward direction; may be
preceded by a plus sign.
Postulate: A statement accepted as true without proof.
Prime factorization: A number written as the product of all its prime factors.
Prime number: A whole number greater than 1 whose only factors are 1 and itself.
Probability: A number (whole, fraction, decimal, or ratio) that shows how likely it is that
an event will happen; chance.
Probability distribution: The set of possible values of a random variable with a
probability assigned to each.
Probability model: A probability model is used to assign probabilities to outcomes of a
chance process by examining the nature of the process. The set of all outcomes is
called the sample space, and their probabilities sum to 1. See uniform probability model.
Product: The answer to a multiplication problem.
Proof: A method of constructing a valid argument, using deductive reasoning.
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Properties of equality: See Table 4.
Properties of inequality: See Table 5.
Properties of operations: A set of rules for determining the order of operations. (See
Table 3)
Proportion: An equation that states that two ratios (fractions) are equal (e.g., 4/8 = 1/2
or 4:8 = 1:2).
Pythagorean Theorem: The square of the hypotenuse is equal to the sum of the
squares of the other two sides (legs): a2 + b2=c2
Q
Quadrant: One-fourth of a coordinate grid, formed by the intersecting axes.
Quadratic equation: An equation that contains a variable raised to the second power;
there may be two solutions to a quadratic equation. Some examples are y = 3x2 - 5x2 +
1, x2 + 5xy + y2 = 1, and 1.6a2 + 5.9a - 3.14 = 0
Quadratic expression: Contains a variable raised to the second power.
Quadratic function: A function that can be represented by an equation of the form y =
ax2 + bx + c, where a, b, and c are arbitrary, but fixed, numbers and a ≠ 0. The graph of
this function is a parabola.
Quadratic polynomial: A polynomial where the highest degree of any of its terms is 2.
Quadrilateral: A polygon with four sides and four angles.
Quotient: Answer to a division problem; the amount in each part of the whole.
R
Radian: A unit of angular measurement such that there are 2 pi radians in a complete
circle. One radian = 180/pi degrees. One radian is approximately 57.3 o.
Radical: The √ symbol, which is used to indicate square roots or nth roots.
Radius: The distance from the middle of the circle to the edge. Radius = 1/2 diameter
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Random: Selected by chance, with no outcome more likely than any other.
Random sampling: A smaller group of people or objects chosen from a larger group
or population by a process giving equal chance of selection to all possible people or
objects.
Random variable: An assignment of a numerical value to each outcome in a sample
space.
Range: The spread between the lowest number and the highest number.
Rate: A ratio of two different kinds of units used to show a relationship:
miles gallon, miles per hour; the percent relationship of the part to the base in a percent
problem.
Ratio: A comparison of two numbers used to show a relationship or pattern.
Rational expression: A quotient of two polynomials with a non-zero denominator.
Rational number: A number expressible in the form a/b or -a/b for some fraction a/b.
The rational numbers include the integers. (See Illustration 1)
Ray: Part of a line having only one endpoint (one side of an angle).
Real number: A number from the set of numbers consisting of all rational and all
irrational numbers. (See Illustration 1)
Rectangle: A four-sided figure in which the opposite sides are equal in length.
Rectangular solid: A three-dimensional figure with six sides, all of which are
rectangles.
Rectilinear figure: A polygon all angles of which are right angles.
Recursive pattern or sequence: A pattern or sequence wherein each successive term
can be computed from some or all of the preceding terms by an algorithmic procedure.
Reflection: A type of transformation that flips points about a line, called the line of
reflection. Taken together, the image and the pre-image have the line of reflection as a
line of symmetry.
Reflex angle: An angle that measures more than 180° but less than 360°
Relative frequency: The empirical counterpart of probability. If an event occurs N'
times in N trials, its relative frequency is N'/N.
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Remainder Theorem: If f(x) is a polynomial in x then the remainder on dividing f(x) by x
− a is f(a).
Repeating decimal: A decimal number that continues infinitely, repeating a pattern of
digits.
Residual: The observed value minus the predicted value. It is the difference of the
results obtained by observation, and by computation from a formula.
Rhombus: A parallelogram with four equal sides.
Right angle: An angle that makes a squared corner that measures exactly 90°.
Rigid motion: A transformation of points in space consisting of a sequence of one or
more translations, reflections, and/or rotations. Rigid motions are here assumed to
preserve distances and angle measures.
Rise: The change in the y value in vertical distance.
Rotation: A type of transformation that turns a figure about a fixed point, also called the
center of rotation.
Rounding: Using an approximation for a number.
Run: The horizontal change in the x value.
S
Sample space: In a probability model for a random process, a list of the individual
outcomes that are to be considered.
SAS congruence: Side-Angle-Side congruence. When two triangles have
corresponding sides and the angles formed by those sides are congruent, the triangles
and congruent.
Scalar: A scalar is a quantity, which has only magnitude but no direction.
Scale factor: The ratio of corresponding lengths of the sides of two similar figures.
Scalene: No equal sides and no equal angles.
Scalene triangle: A triangle in which all three sides have different lengths and all three
angles have different measurements.
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Scatter plot: A graph in the coordinate plane representing a set of bivariate data. For
example, the heights and weights of a group of people could be displayed on a scatter
plot.
Sequence / progression: A set of elements ordered so that they can be labeled with
consecutive positive integers starting with 1 (e.g., 1, 3, 9, 27, 81). In this sequence, 1 is
the first term, 3 is the second term, 9 is the third term, and so on.
Scientific notation: A way of writing very large numbers and very small decimals in
which the numbers are expressed as the product of a number between 1 and 10 and a
power of 10 (e.g., 562 = 5.62 x 102).
Signed numbers: Positive and negative numbers; often used to show quantity,
distance, or direction.
Similar figures: Figures in which the corresponding angles and sides are
proportionate; figures having the same shape, but different sizes.
Similarity transformation: A rigid motion followed by a dilation.
Simple interest: Fee charged for borrowing money (or earned for investing money) for
a particular period of time; simple interest = principal x rate x time.



Principal is the amount of the purchase.
Rate is the percent you will pay for the loan.
Time is how long you are going to borrow the money.
Simplified: Means to reduce to lowest terms; or perform all operations within a
mathematical expression.
Sine: The trigonometric function that for an acute angle is the ratio between the leg
opposite the angle when the angle is considered part of a right triangle and the
hypotenuse.
Slope: Refers to the steepness of the line or rise over run.
Slope-intercept formula: An equation of a line that takes the following form: y = mx +
b, where m is the slope and b is the y-intercept.
Solution set: Not one number but a set of whole numbers.
Solving percents: The base represents the whole amount in a percent problem. The
rate is the percent relationship of the part to the base in a percent problem. The part is a
portion of the whole or base in a percent problem.
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Square: A figure with 4 right angles (a type of rectangle) and 4 sides of equal length (a
special rhombus); numerical operation in which a number is multiplied by itself,
represented by the exponent 2.
Square root: A number is the number that when multiplied by itself equals the given
number.
SSS Congruence: Side-Side-Side Congruence. When two triangles have
corresponding sides that are congruent, the triangles are congruent.
Straight angle: Measures exactly 180°, every straight angle.
Supplementary angles: Two angles for which the sum of their measures is 180°.
Symmetry: The quality of being made up of exactly similar parts facing each other or
around an axis.
T
Tangent: a) Meeting a curve or surface in a single point of a sufficiently small interval
is considered; b) The trigonometric function that, for an acute angle, is the ratio
between the leg opposite the angle and the leg adjacent to the angle when the angle is
considered part of a right triangle.
Tape diagram: A drawing that looks like a segment of tape, used to illustrate number
relationships; also known as a strip diagram, bar model, fraction strip, or length model.
Terms: A single number, a variable, or numbers and variable multiplied together.
Terminating decimal: A decimal is terminating if its repeating digit is 0. A terminating
decimal is the decimal form of a rational number.
Third quartile: For a data set with median M, the third quartile is the median of the data
values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120},
the third quartile is 15. See also: median, first quartile, interquartile range.
Transformation: A prescription, or rule, that sets up a one-to-one correspondence
between the points in a geometric object (the pre-image) and the points in another
geometric object (the image). Reflections, rotations, translations, and dilations are
particular examples of transformations.
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Transitivity principle for indirect measurement: If the length of object A is greater
than the length of object B, and the length of object B is greater than the length of object
C, then the length of object A is greater than the length of object C. This principle
applies to measurement of other quantities as well.
Translation: A type of transformation that moves every point in a graph or geometric
figure by the same distance in the same direction without a change in orientation or
size.
Transversal: A line that crosses two or more parallel lines.
Trapezoid: A quadrilateral with two sides that are parallel.
Triangles: (∆) is the symbol for a triangle. A triangle has three sides and has three
angles, which together add up to 180 and are identified by vertices (points), e.g.,
(∆ABC).
Trigonometric function: A function (as the sine, cosine, tangent, cotangent, secant, or
cosecant) of an arc or angle most simply expressed in terms of the ratios of pairs of
sides of a right angled triangle.
Trigonometry: The study of triangles, with emphasis on calculations involving the
lengths of sides and the measures of angles.
U
Uniform probability model: A probability model which assigns equal probability to all
outcomes. See also: probability model.
Unit fraction: A fraction with a numerator of 1, such as 1/3 or 1/5.
Unit price: The cost of one item.
Unit rate: The rate for one unit of a given quantity; has a denominator of zero.
V
Valid: a) Well-grounded or justifiable; being at once relevant and meaningful (e.g., a
valid theory); b) logically correct.
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Variable: a) Any letter used to stand for a number; b) A quantity that can change or
that may take on different values. It refers to the letter or symbol representing such a
quantity in an expression, equation, inequality, or matrix.
Variable expression: A combination of numbers, variables, and mathematical
operations arranged in a meaningful order.
Vector: A quantity with magnitude and direction in the plane or in space, defined by an
ordered pair or triple of real numbers.
Vertex: The point at which two or more line segments or sides of a figure meet; the
point at which the two rays that form an angle meet.
Vertical angles: The angles across from each other when two lines intersect or cross;
also called opposite angles.
Vertical axis: The scale that runs along the side or top to bottom on a graph or
coordinate grid (the y- axis).
Visual fraction model: A tape diagram, number line diagram, or area model.
Volume: A measure of the amount of space inside a three-dimensional or solid figure.
Volume is always measured in cubic (3) units.
W
Whole numbers: The numbers 0, 1, 2, 3,… (See Illustration 1)
X
(x,y) coordinates: An ordered pair; enclosed in parentheses, separated by a comma.
x-axis: The horizontal axis in a coordinate graph.
x-coordinate: The first number in an ordered pair; the distance from the origin along
the x-axis.
x-intercept: The point at which a line crosses the x-axis on a coordinate graph; the
ordered pair (x,0).
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Y
y-axis: The vertical axis in a coordinate graph.
y-coordinate: The second number in an ordered pair, the distance from the origin
along the y-axis.
y-intercept: The point at which a line crosses the y-axis on a coordinate graph; the
ordered pair (0,y).
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Tables and Illustrations of Key Mathematical Properties,
Rules, and Number Sets
TABLE 1. Common addition and subtraction situations.21
Add to
Take from
Put
Together/
Take
Apart23
Result Unknown
Two bunnies sat on the
grass. Three more
bunnies hopped there.
How many bunnies are
on the grass now?
2+3=?
Five apples were on the
table. I ate two apples.
How many apples are
on the table now?
5–2=?
Change Unknown
Two bunnies were
sitting on the grass.
Some more bunnies
hopped there. Then
there were five
bunnies. How many
bunnies hopped over
to the first two?
2+?=5
Five apples were on
the table. I ate some
apples. Then there
were three apples.
How many apples did
I eat?
5–?=3
Total Unknown
Addend Unknown
Three red apples and
two green apples are on
the table. How many
apples are on the table?
3+2=?
Five apples are on
the table. Three are
red and the rest are
green. How many
apples are green?
3 + ? = 5, 5 – 3 = ?
Start Unknown
Some bunnies were
sitting on the grass.
Three more bunnies
hopped there. Then
there were five
bunnies. How many
bunnies were on the
grass before?
?+3=5
Some apples were
on the table. I ate
two apples. Then
there were three
apples. How many
apples were on the
table before?
?–2=3
Both Addends
Unknown22
Grandma has five
flowers. How many
can she put in her
red vase and how
many in her blue
vase?
5 = 0 + 5, 5 = 5 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
21
Adapted from Boxes 2–4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32–33).
These take apart situations can be used to show all the decompositions of a given number. The associated
equations, which have the total on the left of the equal sign, help children understand that the = sign does not
always mean makes or results in but always does mean is the same number as.
23
Either addend can be unknown, so there are three variations of these problem situations. Both Addends
Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10.
22
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Difference Unknown
(“How many more?”
version):
Lucy has two apples.
Julie has five apples.
How many more apples
does Julie have than
Lucy?
Compare24
(“How many fewer?”
version):
Lucy has two apples.
Julie has five apples.
How many fewer apples
does Lucy have than
Julie?
2 + ? = 5, 5 – 2 = ?
Bigger Unknown
(Version with “more”):
Julie has three more
apples than Lucy.
Lucy has two apples.
How many apples
does Julie have?
(Version with “fewer”):
Lucy has 3 fewer
apples than Julie.
Lucy has two apples.
How many apples
does Julie have?
2 + 3 = ?, 3 + 2 = ?
Smaller Unknown
(Version with
“more”):
Julie has three more
apples than Lucy.
Julie has five apples.
How many apples
does Lucy have?
(Version with
“fewer”):
Lucy has 3 fewer
apples than Julie.
Julie has five apples.
How many apples
does Lucy have?
5 – 3 = ?, ? + 3 = 5
24
For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version
using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.
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TABLE 2. Common multiplication and division situations.25
Unknown Product
Group Size Unknown
Number of Groups
Unknown
(“How many in each
group?” Division)
(“How many groups?”
Division)
Equal Groups
Arrays,26 Area27
36=?
3  ? = 18 and 18 ÷ 3 = ?
?  6 = 18 and 18 ÷ 6 = ?
There are 3 bags with 6
plums in each bag. How
many plums are there in
all?
If 18 plums are shared
equally into 3 bags, then
how many plums will be
in each bag?
If 18 plums are to be
packed 6 to a bag, then
how many bags are
needed?
Measurement example.
You need 3 lengths of
string, each 6 inches long.
How much string will you
need altogether?
Measurement example.
You have 18 inches of
string, which you will cut
into 3 equal pieces. How
long will each piece of
string be?
Measurement example.
You have 18 inches of
string, which you will cut
into pieces that are 6
inches long. How many
pieces of string will you
have?
There are 3 rows of
apples with 6 apples in
each row. How many
apples are there?
If 18 apples are arranged
into 3 equal rows, how
many apples will be in
each row?
If 18 apples are arranged
into equal rows of 6
apples, how many rows
will there be?
Area example. What is the
area of a 3 cm by 6 cm
rectangle?
Area example. A rectangle
has area 18 square
centimeters. If one side is
3 cm long, how long is a
side next to it?
Area example. A rectangle
has area 18 square
centimeters. If one side is
6 cm long, how long is a
side next to it?
25
The first examples in each cell are examples of discrete things. These are easier for students and should be given
before the measurement examples.
26
The language in the array examples shows the easiest form of array problems. A harder form is to use the terms
rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there?
Both forms are valuable.
27
Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array
problems include these especially important measurement situations.
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A blue hat costs $6. A red
hat costs 3 times as much
as the blue hat. How
much does the red hat
cost?
A red hat costs $18 and
that is 3 times as much as
a blue hat costs. How
much does a blue hat
cost?
A red hat costs $18 and a
blue hat costs $6. How
many times as much does
the red hat cost as the
blue hat?
Compare
Measurement example. A
rubber band is 6 cm long.
How long will the rubber
band be when it is
stretched to be 3 times as
long?
Measurement example. A
rubber band is stretched
to be 18 cm long and that
is 3 times as long as it was
at first. How long was the
rubber band at first?
Measurement example. A
rubber band was 6 cm
long at first. Now it is
stretched to be 18 cm
long. How many times as
long is the rubber band
now as it was at first?
General
ab=?
a  ? = p and p  a = ?
?  b = p and p  b = ?
TABLE 3. The properties of operations.
Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply
to the rational number system, the real number system, and the complex number system.
Associative property of addition
Commutative property of addition
(a + b) + c = a + (b + c)
a+b=b+a
Additive identity property of 0
a+0=0+a=a
Existence of additive inverses
For every a there exists –a so that a + (–a) = (–a) + a = 0.
Associative property of multiplication
Commutative property of multiplication
(a  b)  c = a  (b  c)
ab=ba
Multiplicative identity property of 1
a1=1a=a
Existence of multiplicative inverses
For every a  0 there exists 1/a so that a  1/a = 1/a  a = 1.
Distributive property of multiplication
over addition
a  (b + c) = a  b + a  c
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TABLE 4. The properties of equality.
Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.
Reflexive property of equality
a=a
Symmetric property of equality
If a = b, then b = a.
Transitive property of equality
If a = b and b = c, then a = c.
Addition property of equality
If a = b, then a + c = b + c.
Subtraction property of equality
If a = b, then a – c = b – c.
Multiplication property of equality
If a = b, then a × c = b × c.
Division property of equality
If a = b and c ≠ 0, then a ÷ c = b ÷ c.
Substitution property of equality
If a = b, then b may be substituted for
a in any expression containing a.
TABLE 5. The properties of inequality.
Here a, b, and c stand for arbitrary numbers in the rational or real number systems.
Exactly one of the following is true: a < b, a = b, a > b.
If a > b and b > c then a > c.
If a > b, then b < a.
If a > b, then –a < –b.
If a > b, then a ± c > b ± c.
If a > b and c > 0, then a × c > b × c.
If a > b and c < 0, then a × c < b × c.
If a > b and c > 0, then a ÷ c > b ÷ c.
If a > b and c < 0, then a ÷ c < b ÷ c.
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ILLUSTRATION 1. The Number System.
The Number System is comprised of number sets beginning with the Counting Numbers and culminating
in the more complete Complex Numbers. The name of each set is written on the boundary of the set,
indicating that each increasing oval encompasses the sets contained within. Note that the Real Number
Set is comprised of two parts: Rational Numbers and Irrational Numbers.
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Appendix A – Financial Literacy Resources
National Standards for Adult Financial Literacy Education
(www.financiallit.org/resources/standards.aspx)
The National Standards for Adult Financial Literacy Education identify the personal
finance knowledge and skills an adult should possess. The Institute for Financial
Literacy® contends that all adults who receive financial literacy education should have,
at a minimum, the knowledge and ability to competently perform the basic tasks of
managing their personal finances.
Practical Money Skills for Life (www.practicalmoneyskills.com)
To help consumers and students of all ages learn the essentials of personal finance,
Visa has partnered with leading consumer advocates, educators, and financial
institutions to develop the Practical Money Skills program. At this website you can
access free educational resources, including personal finance articles, games and
lesson plans.
Money Smart for Young Adults (www.fdic.gov)
The FDIC’s Money Smart for Young Adults curriculum helps youth ages 12-20 learn the
basics of handling their money and finances, including how to create positive
relationships with financial institutions. Equipping young people in their formative years
with the basics of financial education can give them the knowledge, skills, and
confidence they need to manage their finances once they enter the real world. Money
Smart for Young Adults consists of eight instructor-led modules. Each module includes
a fully scripted instructor guide, participant guide, and overhead slides. The materials
also include an optional computer-based scenario that allows students to complete
realistic exercises based on each module.
Money Skill (www.moneyskill.org)
MoneySKILL is a free online reality based personal finance course for young adults
developed by the AFSA Education Foundation. This interactive curriculum is aimed at
the millions of high school and college students who graduate each year without a basic
understanding of money management fundamentals. The course is designed to be used
as all or part of a grade for courses in economics, math, social studies or where
personal finance are taught. Students experience the interactive curriculum as both
written text and audio narration. In addition, frequent quizzes test their grasp of each
and every concept. The 36-module curriculum, with pre- and post -tests, covers the
content areas of income, expenses, assets, liabilities and risk management. A life
simulation module asks students to project their own financial life expectancies in areas
such as employment, housing, transportation, education, marriage, family and
retirement. The life simulation allows students to incorporate the MoneySKILL personal
finance concepts into their everyday lives, thus providing them with knowledge and
skills that will last a lifetime.
More financial literacy resources available at: sabes.org/resources/financial-literacy.htm
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